Optics (Lec - 26) - Electronics & Communication Engineering Video Lecture - Electronics and Communication Engineering (ECE)

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okay first of all I was in touch by
email with George and he asked me to
flash this up again which which was from
a couple of lectures ago because he
thought this was very critical and very
important that you sort of understood
what it says so I'm going to just say a
few things about it I guess the first
thing to note is that it it compares
these two different cases coherent
imaging and incoherent imaging and you
see that apart from that these two the
top half and the bottom half are really
identical so you use the same method in
in either case except that of course in
the coherent case you're working with
amplitudes whereas in the incoherent
case you're working with intensities
okay but apart from that it's identical
so what it's saying is if you illuminate
the thin transparency then you get a
field on the far side of this
transparency which goes into your into
your optical system so this is his Gian
and to calculate the image you can
either do it by convolving that with a
point spread function to get the
amplitude out or you can do it in the
Fourier domain by Fourier transforming
to get the object spectrum and then
multiplying it by a transfer function
and then getting the the field out okay
and then once you've got the field of
course you can calculate the intensity
from that just by doing the modulus
square so the difference between these
two things is here we're working with
amplitudes so we have a coherent point
spread function and an amplitude trans
trans transfer function whereas furring
this one the in coherent case we have an
intensity point spread function which is
just the
modular square of this and here we have
an optical transfer function which is
the autocorrelation of this okay so so
that's really all there is to to do or
there is that you need to know about
calculating images ok any questions
about that so I'm just as I saw just
flash that because George said he wanted
me to and now we'll go on to where we
got up to last time in the lecture so
George suggested he sent me some more
material for today and he also asked me
to add some extra to it of my own
material but I don't know how far we get
with this the so we've got still got
some left over from last time actually
and then the bit that George sent you
might be fed up by then but it anyway if
we've still got some time and you're not
fed up I'll carry on with some stuff
that I put in there which is to do with
focusing and but it might be you might
think it said I sort of rather higher
level than the rest of the course so you
know this is not for that this bit that
I've added is is not for examining or
anything it's it's it's meant just for
your interest anyway this is where we
got up to the defocus RACF so there's
this again you remember let's make it
bigger what's going on I'm a I've gone
the wrong thing I should that sorry I've
got my glasses on I can't see what I'm
doing and it's all in a different place
from of course the the version of PDF
that for the PowerPoint that I use
anyway here we are
the the D focus ATF so you remember we
introduced this last time we said if
you've got
the focused objects then you can work
out the imaging of it by using a an
amplitude transfer function just like
before but it's not just the normal one
it's modified by the effects of defocus
what happens is that you have to
multiply the transfer function by a
complex exponential e to the I this
thing it therefore becomes complex and
you can divide that up into two parts a
real part and an imaginary part the real
part will image the real information of
the objects which is basically the
absorption of the objects and that's
what this shows here the real part of
that transfer function which is cosine
of this thing and so this is the cutoff
of the optical system here sorry no it
isn't is it no sorry the cutoff I guess
is some arbitrary place I guess it's not
actually shown here but anyway the the
fact is that as you change the amount of
defocus this will be scaled accordingly
relative to the cutoff of the of the
transfer function right so you can
imagine if the transfer function was
right in here you it would be almost as
though it was the same as an in focus
system and then as you as you make the
the cutoff wider and wider relative to
this cos curve you'll see you'll get
more and more of these Wiggles so this
was a case where it's not changing very
much so there we are
there's the region where it's nonzero
and and so it's not changing very much
in that region and so we said that
corresponds to mild D focus and and then
the next case we looked at was was now
when it's got a lot of D focus all right
so you can see here it starts really
Wiggly
remember I said this last time this is
actually cause of this thing squared so
that's why it doesn't look like an
ordinary cos function because of that
the x squared so it scales it in this
nonlinear way and so here's the cutoff
again in this case now in this case
you've got some Wiggles and those
Wiggles are going to lead to artifacts
in the image
there's firstly because of the fact that
this goes negative you can see that some
spatial frequencies will be imaged with
the wrong sign so it's just like you
know if you've got a a Fourier series
for let's say a square wave if you take
the Fourier series for a square wave and
you and you change the sign of some of
those components it doesn't look like a
square wave anymore right so you you get
a very poor image in that case so this
is really bad news for the image and and
the other thing that George mentions up
there is that of course there's regions
where the value of the transfer function
is very small so this spatial frequency
here freeze them for example is not
going to be imaged at all so you'll get
some gaps in the in the response of the
different transfer of spatial
frequencies okay so the significance of
depth of focus in imaging so here he
shows a 4f system it shows here the NA
in and the Na out and you remember of
course that the as you change the na you
also change the magnification you change
the lateral magnification and you also
change the longitudinal or axial
magnification that goes as the na
squared and so we define these two terms
depth of field is basically the the D
focus tolerance in the object space how
far you can move the objects without it
going too much out of focus
and so this gives some some idea about
the the value of that then the depths
that depth of field is given by this
lambda over 2 to na squared right
because the the you get this na squared
because the longitudinal magnification
also goes as na squared and then on the
other side in the image side you have a
depth of focus rather than a depth of
field notice in the title he calls it
DOF which can mean either so depth of
field depth of focus you'll find that of
course lots in general terminology
people often use these phrases almost
interchangeably and are not very careful
about using the right one for the right
for the right space but this is what it
really means depth of field is how much
you can move the object depth of focus
is how much you can move the screen here
without it going out of focus right and
you can see that the this is expressing
the depth of focus now in terms of
numerical aperture you get the same
expression but it's now this n a rather
than that one and a couple of course the
magnification also goes as na squared as
well so so the the image of the bitless
in focus here of the objects is of
course the same as the depth of focus of
the system all right they're all clear
yeah good ok
yeah even though our derivation were
character carried out for spatially
coherent imaging exactly the same
arguments you can use for the incoherent
case right so the idea of a we showed
that the focused amplitude transfer
function in the same way you can also
have a D focused OTF
and right so this what let's think of
now what happens if we've got some
objects which has got some height
variations here he's taking in the case
of a surface and he's called this a two
and a half the object I think he's
probably called it that because I
sometimes use that terminology I'm not
the only one really I didn't invent it
at all but people often use this two and
a half D to represent something which is
not a 2d object it's not flat it's got
some depth but it's not a true 3d
objects in that it's got no internal
structure it's just a surface right so
anyway in this case this is the object
and this is its image and you can see
that if if this object is quite deep
it's going to be deeper than the depth
of field of the system and so some parts
of that are going to be out of focus and
so this is a region for example which is
a long way away from the focal plane and
therefore it's going to be imaged out of
focus and here here it corresponds to
this region here and notice if you look
at the this this image it's inverted in
X but a distance you see that in in the
Z Direction a distance which is move
this way here is also moved this way
here so there's no inversion in the in
the Z direction I did I hear it's a
question over there or is it just
someone shuffling around no okay okay
so as George points out here what this
means then for this particular object is
the point spread function is not now
spatially invariant right it's a
spatially a shift variance PSF because
different parts of this surface are
going to experience a different point
spread function because they might be in
or out
focus so yeah and the under part just
before like before at the part which is
in focus is given by this depth of field
which we just had before right so you
can either think of it as a depth of
field or a depth of focus the same
expression right so this is an example
of this let's take an object which is a
letter M I guess he didn't use MIT this
time because he wants to use the same
for each so that you can see how how
each of them is affected by the
different point spread function so this
is in the focal plane and then this is
to depth of fields out of the focal
plane this is for depth of fields out of
the focal plane so when you image that
you can see that it's going to become
this one's going to be in focus it's
still a bit blurred of course because of
the the diffraction effects in the
imaging so this is convolved with the
point spread function of the coherent
optical system and and then this is for
the the the two a defocused ones getting
more and more out of focus so right now
once we've got that blur what can we do
about it and there's been quite a lot of
papers that have looked at this question
of whether you can actually undo the
blur d blurring they call it or
sometimes deconvolution because if you
think of the blurring as being a
convolution you want to do undo that you
want to do a deconvolution so what we
actually get there in the in the Fourier
domain the spectrum of the image is
equal to the spectrum of the object
multiplied by some transfer function and
this will be true whether it's coherent
or incoherent in general there'd be the
same sort of
facts but you'd have to work with
intensities rather than add virtues and
so what we say then is that if we this
is the blurred thing so we can actually
do blur it by just dividing by that to
get rid of it so if we get our image we
multiply by one over the transfer
function and that gives us our D blurred
version of the of the image so as he
calls it here you can record it inverse
filtering or sometimes called
deconvolution but that is actually you
know you probably you all thought of
that can work as it sounds they ought to
work and of course it doesn't work very
well we're going to show a bit more
about this in a minute
in particular it really goes wrong if
this thing goes to 0 obviously so you
can't do very much about that and and
then also if it goes to something which
is even very small you have to amplify
it by a lot and when you amplify you're
going to amplify the noise so you'll end
up with a noisy reconstruction after
you've done that right so so this sort
of inverse filtering invariably
increases the noise in the image and so
how can we get around that well we just
get around the zero by making making
making it so that there isn't a zero
anymore so this is our new version and
you can see that if if moon was 0 this
would just become the same as before
this is H times H star so the H stars
will cancel and it's 1 over H right so
so if if if this is big compared with
that then it becomes the same as before
but if this becomes small then we're
just left with this and it avoids it
blowing up and giving a infinite
so yeah so if mu is not it reduces to
the direct filter and we just described
that that's not a good thing to do so
the more noise there is there the the
bigger you need to make this to avoid
getting too much not noise in the in the
image so okay so as the noise increases
you would tend to increase the value of
mu yeah so here it's saying that if the
signal are noise both of a particular
type of statistics then you can have you
can find what the optimum value of MU is
and we find then that it's equal to 1
over the signal-to-noise ratio and and
this is what's called a vena filter so
this that this n Ruggiero tree comes
down to be the same as of veena filter
which you might have come across some of
you in in other things right so so all
you're doing really is you're amplifying
the spatial frequencies that have been
pushed down by the imaging process but
if there's if they become smaller than
the noise then you don't do that anymore
right so he's going to give an example
of this so so this is the convolution
using this chicken of regular lized
inverse filter and yeah okay he's saying
here that he's assuming he's assuming he
knows the depth before he does this
right so obviously that is something you
wouldn't always necessarily know but he
knows the depth therefore you can
calculate what the what the transfer
function was on the imaging
died and therefore he knows what filter
to inverse filter to apply in order to
get the best reconstruction right and
you can see that it works extremely well
for this one this one is slightly
blurred slightly a bit more blurred and
you can see also that the background
here or at least I can see on this
screen but this one doesn't look very
much the background is becoming not
black anymore okay and this is what
happens if you've got a noisy image if
you add noise to the to the original to
the image before you do the
deconvolution now you see that that
actually it's not working nearly so well
and this one here you can't really make
out what there is there right so this is
showing that that these methods can work
but you can see that there are also
limits to how well they can work and
especially in the presence of noise
actually I should mention that in of
course in MIT in the Media Lab there's a
lot of work going on on in this area of
what they call computational photography
which is all to do with changing the the
properties of the camera in order to
make it so that this sort of
reconstruction it is more efficient so
it turns out that if you're clever about
the design of the imaging system you can
actually make it so that this all works
a lot better but we weren't going into
all that I'm sorry that I've got this
wrong this one was the sorry this was
the oh yeah that's right this is the
image before you've done the inverse
filtering and this is the image after
you've done the inverse filtering so
it's actually doing quite well this one
here is is pretty well not very
different from that and this one as
before is is not as good but but doing
pretty
pretty well okay of course if you didn't
know what the the the height of these
things is then you might have to use
some iterative algorithm in order to
come up with the the best reconstruction
you can imagine also that so this is for
this so-called two and a half the object
you could imagine an object which really
did have structure inside itself so
maybe these three M's would be on top of
each other and semi-transparent and and
then that the trick would be to try to
actually produce a reconstruction of the
3d object from the 3d image and there
are there are actually computer packages
that you can get to do just that very
sort of thing okay so that was that was
what we should have done and we now go
on to today's lecture and so today's
lecture is about first of all about
polarization and which we've been
talking about in the lab today so
yesterday wasn't it so that's a bit
topical and then George wrote these so
so the bit that I've added extra is
basically addressing these points here
Effects of polarization on imaging in a
microscope objective or a lens with a
high numerical aperture so we start off
with the polarization I'm not sure what
you said about polarization first of all
but basically then the wave equation for
in the electromagnetic field is
basically a vector wave equation it's a
function of electric field here and and
we had then that the polarization is
some function of the electric field
and how the relationship between these
depends on the sort of material and so
this is giving some different examples
the simple case is this one which which
we did look at earlier is where the
polarization is proportional to the
electric field and you remember that
that D you get D by adding the piece of
the e and so here this is called the
electric susceptibility and P equals KY
times e so if this is a constant this is
just a linear relationship which is
where how we described it before but
there are two ways that that can break
down in practice and usually actually
for materials where it breaks down it
breaks down for both at the same time
unfortunately so it becomes very
complicated but here we've written it
with these two effects separately and
this is this is where actually the
relationship is still linear but P is
related to e naught by a simple scalar
relationship like this but by a tensor
relationship so the direction of P is
not in the same direction as the
direction of e and so you've got this
three by three matrix which describes
the properties of the material right and
this will depend on the on well for a
isotropic material like a crystal or
something like that it would depend on
the crystal structure of course if
you've got some complicated varying
object then it can be all sorts of even
more complicated things and so that's
the that's the the first case so that
this is the way what this is what's
going to lead to some interesting
polarization effects as you put that
electric field into a crystal for
example and then this is the other one
which is now looking at an isotropic
material but non linear so this is now
longer a constant and so what that means
you can you could expand it as a power
series and you can see that there might
be some extra terms here this is what's
called thicker effects there's
effectively a susceptibility which is
which is dependent on the intensity of
the light which shines on it and and of
course you might expect that there will
be higher orders of this as well of
course that these sorts of effects
you're only going to excite these if you
if you have a large value of AE you know
it's a bit like if you imagine how this
all comes about this comes about because
your material is made up of atoms which
are all bonded together and joined to
each other with Springs
right and when you apply the electric
field it makes these atoms vibrate
against the spring and if the vibrations
are small then then it's going to
satisfy Hookes law and it's going to be
linear but if you increase the forcing
term then the vibrations get bigger
Hookes law breaks down and the and the
vibrations become nonlinear that's the
basic physics of what's going on here
okay so so this is what's called the
Kerr if X there is actually another
effect which george hasn't exist hasn't
mentioned here there's another you see
this is actually e cubed here but you
can't have an effect where this is e
squared which is called the what's it
called the pockels effect I was really
only asking because I had forgotten the
word for a minute but my students here
know that I'm always forgetting names
and so I they usually come back
eventually but did you see that what
that did ah yeah it did this okay the
relationship between the SEC
susceptibility and the refractive index
and and this is saying that the the
phase delay depends on the polarization
right and and this one the index of
refraction depends on the intensity all
right so this is nonlinear optics and as
I say those last two effects very often
come together right so this is now
talking about polarization of light I
think that probably George must have
mentioned this to some degree earlier on
but this is what this is showing then is
a plane wave of light it's linearly
polarized in the X direction sorry the Y
direction this is taken as Y so the
electric field is in the in the Y
direction which actually means that the
magnetic field is going to be in the X
direction because they're at right
angles with each arrow if you remember
and you remember that so you can think
of the wave propagating in two different
ways one is to think of it at a certain
instant of time so what we call here
it's a snapshot and then we plot what
happens as a function of Z so this is of
course we have to do this because it's
difficult to plot it against space and
time at the same time because difficult
to visualize what's going on so you can
see here this electric field is always
in the same plane and it's got a it's a
maximum here it reduces it becomes zero
it then increases in the negative
direction and comes back to zero again
and so on the side right so so this is
linearly polarized light as a function
of space and then this one shows the
same thing as a function of time so
we're looking at now at one particular
value of positions N equals naught and
we're looking at how this varies as a
function of time exactly the same
diagram of course you probably saw when
I
when I had flicked their to the next
slide it didn't change at all it just
changed the the caption and this is what
circular polarization does circular
polarization you can see what this is
trying to it looks a bit complicated
here but what it's really doing is
actually quite straightforward it's
actually rotating this electric vector
is rotating either with distance or with
time according to which way you do the
plot and the length of this line stays
the same so this is what we mean by
circular polarization and this is
showing how actually you can think of
circular polarization has been made up
of two components one linearly polarized
in this dial in the Y Direction one
linearly polarized in the X direction
and you've so you can see that they're
they're also out of phase by 90 degrees
so you can see now that in at this
distance Y a position this is plotted
against said again here this one is a
maximum this one is zero so the electric
field is in the X direction and then as
you go as you travel along this gets
smaller but this gets bigger so at this
position here the electric field is now
polarized in the Y direction and then it
got this one goes back down to zero
again and this one gets bigger right so
so this is exactly equivalent to the
case that I showed before the circular
polarized case I might add the
elliptical polarization is where the
amplitude of this component and this
component are not equal so if that was
the case it would gain you that you
think of this electric field vector is
rotating but the length of the vector
would also change as it rotates and of
course you can have two cases you can
have it rotating
clockwise or you can have it rotating
anti-clockwise so these are called
left and right hand circular
polarization okay so so this is showing
like an end on view so this is it this
is x and y so this thing is is pointing
out towards you isn't it that means no
it means no it means it's going away
isn't it but is it right because that's
it's it's it's a left handed coordinate
system if that's the case because X oh
no it's not no it's not no it's right
it's right it's a right handed
coordinate system okay yeah x cot X
cross Y goes away from you right and so
this is showing then end on what happens
to this electric vector and the wonders
of modern science of in fact in fact
changed the circular polarization to
electrical polarization without me doing
anything right now so next is to
introduce some sorry that's I missed
that one there were lots of these things
that my computer misinterpreted from
Georgie's computer and I changed most of
them but I miss one there that should
say lambda over for plate I don't know
what the difference is what he put in
there do you know any did you know when
he's put in there that make it go like
that anyway he did this on his Mac so
yeah yeah but it's a funny pond as well
anyway and so this is showing a what
would happen if you put some
birefringence a special birefringence
material here which is what's called a
lambda by 4 plate and what this does is
it changes the relative phase of the
components of electric electric field in
two directions by a different
mount as it transmits through the
through the crystal and so if you if you
arrange the orientation of it correctly
relative to the light it has this effect
of changing the linear polarization into
circular polarization the way it really
does that is that this you've said that
this you can think of as being made up
of two linearly polarized components and
this you can always think of as being
made up of two linearly polarized
components as well 45 degrees so that so
you just resolve that into two
components right so the circular
polarized you call you that you get that
very simply by just changing the
relative phase of those two linear
polarized components that's how it works
but I think here he's just trying to
show you what it does and then the other
important device that people use is a
lambda by 2 plate and rather than a lamb
but that lambda by 4 plate and that has
the property of changing X polarized
then a linearly polarized light into Y
polarized light and again you can see
how this is working by thinking in terms
of resolving this into two components at
45 degrees to it and then changing the
relative sign maybe I should do that
and this doesn't because this is it
switched on so maybe I can just do that
quickly so what I'm saying is that this
you can think of it this you this this
one you can think of as being resolved
into these two components and then the
crystal is then changing the relative
phase of one of these relative to that
making that and so now the polarization
is like this right so so that's how it's
working for the lambda by 4 plate what
it
during was it was changing the relative
phase of this and this by 90 degrees not
via 180 degrees which would give you a
circular polarized life but in order to
get this effect you have to get the
right orientation of the of the
polarization relative to the the crystal
axes so that that part he hasn't
actually said anything about here right
the other thing maybe you'd be
interested to know this of course if you
put linearly polarized light in and get
circularly polarized light out what
would happen if you went the other way
well in fact it does work the other way
you can put in circularly polarized
light and get out linearly polarized
light
alright so sorry you can you can do
either of those two things the other
thing that you might worry about is it's
interesting that if we change if we
change the phase of this light then it's
going to change the phase of that light
isn't it and yeah and maybe less
I'm not leaving anywhere oh it's any
more than that okay think about this
okay yeah oh well there we saw I've
already jumped the gun a bit so he's
saying what happens here then so we've
got incoming linearly polarized light
the lambda by 4 plate what does the
lambda by 4 plate do circular polarized
so we've got circular polarized light on
here it strikes a mirror and it reflects
what what do we get after its reflected
that's a bit hard now isn't it opposite
circular right and then this gets back
to here and it goes back through this
birefringence plate so what what does it
do same polarization is
right it rotates 500 800 and 80 degrees
okay it's as though actually had gone
through a lambda by 2 plate then isn't
it
well I mean
oh yeah yeah sorry yeah you're quite
right yeah yeah okay yeah okay wait
everyone about that I guess the next
thing is to say what would happen if you
change that mirror by I phase conjugate
mirror but maybe that's too difficult
for us to think about okay and then
remember that I think we had this before
as well that when you when you have
lights interfering the electric field
it's only the parallel components of
electric field that interfere so if
you've got two waves where the electric
fields are at right angles to each other
then they don't interfere so this is
showing this case here we've got two
plane polarized waves that they're
polarized right angles to each other and
so they won't interfere and therefore if
you look at the try to get an
interference pattern you won't see
anything in this case right so and and
this is showing them in in in the same
plane and then they would interfere and
you would get some interference pattern
with with maxima and minima like this
right another interesting one I guess I
might draw over here is what happens if
you this is our screen and let's say
we've got two waves coming in one let's
say is polarize like this and the other
one is coming in here and is polarized
like this you see so we've got these two
waves coming in and you can see that
their electric field vectors are at
right angles to each other and so this
is another example where they're not
going to interfere right so you wouldn't
see any interference fringes in this
case so this corresponds to what's
called P polarization so if we did this
the opposite if we did it with the
electric field in this direction and the
other ones coming in
also with the electric field in this
direction then you can see that these
these are now in parallel directions and
so you would see fringes where these two
plane waves interfere right so I've
drawn this for the particular case this
is supposed to be 90 degrees right
right okay um that's oh that's all
Georgie's material that was the end of
George it didn't end with a particularly
startling thing but so that's that
that's that's so really the end of the
the the course as George wrote it and
but he did ask me as I say to do a bit
more so what I think I do why don't we
have a break now for five minutes and
then I'll start again in five minutes
time so this is now going to be talking
about focusing of light and I said
before the this material I'm going to do
now is just for interest it's not going
to be examinable and if you're not
interested you can go and have a cup of
coffee instead I don't really mind but
some people might be interested in
learning a bit more about focusing of
life
right okay so I was going to say a bit
more about focusing of light and in
particular some of the funny things that
happen when you look at focusing of
light with a microscope objective so the
tight focusing of light and this is
actually a picture of the intensity in
the focal region of a lens not high
numerical metin2 high numerical aperture
in this case this is according to
praxeological taken from born and Wolfe
and it's it's similar but not identical
to the one that George showed on the
last lecture or the lecture before and
so this is the focal plane this is the
optic axis and of course what you're
really seeing this is actually going to
be radially symmetric so if you imagine
this as being revolved around the axis
you'd have the the shape of this
complete surface and you can see what
happens is so this is the focal plane
where you've got the airy disc and then
as you go out of focus then you start
getting this strange sort of almost like
a sort of butterfly shape and yeah so
this calculation that's given in Bourne
and Wolfe
it first of all as I've said it's it's
using this small angle approximation us
assume that sine theta is the same as
theta and it also makes an assumption
which is called the Debye approximation
and so I'll say a bit more about the
Debye approximation a bit before I carry
on I would say another thing I'd
forgotten I'd put this in and that is
that there was a really neat paper
published way back in 1964 by McCutcheon
where he showed
that you can postulate the idea the
concepts of what he calls the 3d pupil
and this is like a cap of a sphere where
it's cut off with this at this angle
alpha the angle of the lens and if you
do love the 3d Fourier transform of this
Dada you get back to that so isn't that
neat so I guess if you think a bit more
about it it's probably I think you can
satisfy yourself that perhaps it's right
this is in case space effectively then
and what it's really saying is that the
because the the light satisfies the wave
equation and the modulus of K is
constant and therefore it's the locus of
this is the is is a sphere of radius K
right so very simple so so like
propagating this convergent light
propagating will have a K vector which
lies on the surface of this sphere so I
just added that just because I thought
people might find it a bit intriguing it
you can sort of generalize a lot of
imaging theory into 3d using this same
sort of approach but I'm not going to
say anything more about that now and
what I'm going to do now is to talk
about yeah that so this is this Debye
approximation again you remember we said
there's two ways of calculating to two
different optical systems which give
virtually the same or similar sort of
results this one I look at first of all
this is where you've got the aperture
stop in the front focal plane of the
lens this is a distance F this is a
distance F and then in this plane here
you will get the airy disc right so the
pattern I described a couple of slides
ago for the field in 3d space would be
in this region here so in this place
here you get Fran off a diffraction and
then in the regions around it you get
for an El diffraction but you can get
quite similar results if you remember by
using this sort of system so this is
where the the aperture stop is not
actually in the front focal plane of the
lens but it actually in the plane of the
lens or you could think another way of
doing this experiment would be to get a
convergent beam and place a circular
aperture in that convergent beam so the
question is what do you get in the focal
region in that case and it turns out
that if you if you satisfy this
condition that a squared over lambda Zed
is very large where as a is a is this
radius here this is called the Fran L
number and if that quantity is very
large it means that you what you'll get
in this focal region is very similar to
this actually there's a parabolic phase
term that we described when we did the
Fourier optics that makes it different
but the intensity is going to be the
same as for this case here right so
these are two cases where you can get
this to by approximation so in this case
it's an its exact for this particular
geometry in this case it's an
approximation which is obeyed if this
pronoun amber is very large and and then
not that long ago people realized that
this was an approximation an assumption
that was made and there was a very nice
paper published by Liam Wolfe so this
paper here 1984 are quite a long time
ago now but not that long but anyway as
tight as as history goes
Liam Wolfe and they they showed what
would happen if this for now
was not large but was of some
intermediate value in this case N equals
10 and so you see what happens
is that this pattern has got distorted
you can see that it's tending to sort of
fan out and the physical explanation of
that is that effectively the light is
being deflected by the aperture and
spreading because of the because of
diffraction effects and that is
superimposed on the focusing effect so
they the lens is trying to focus the
light down and the diffraction is
tending to focus the light out and so
you get a sort of balance between these
two things and so in this focal plane
here you get the same same intensity as
you would if the Debye approximation was
was valid but in the region elsewhere
it's a bit different it's scaled
according to some some sort of a fine
transformation I think it probably would
be or do you think what do you drink
salad and but one it won't very
interesting thing that happens is that
it turns out you can't really see it
very well from this diagram but the
maximum intense intensity is no longer
in the focal plane no longer in the
geometrical focal plane it shifted
towards the aperture and that is called
the focal shift effect and there's been
a lot of papers that have been concerned
with that and calculated what happens
for lots of different cases right so
that's the Debye approximation that
shows what happens if it isn't true
but now I'm going to go back and all the
rest I'm going to carry on now is
talking about what happens if the
paraxial approximation breaks down and
but the Debye approximation is valid the
case when that the by approximation is
not valid and it's a high numerical
aperture is very complicated
there have been very few papers that
have treated that case anyway tight
focusing of light we get light we put it
into a microscope objective the light
then you know a microscope objective a
dry microscope objective might have a
numerical aperture of 0.95 which means
that the light is converging something
maybe over 70 degrees relative to the
axis so very you know you can't really
think of 70 degrees as being a small
angle I guess
so the normal Praxian approximations
break down and this is some sort of
areas where we might want to do this the
one of course that I'm most interested
in microscopy but also laser
micromachining optical data storage
optical lithography laser trapping and
cooling looking at the physics of light
atom interactions and a cavity in
quantum electrodynamics so there's a lot
of basic physics that's being done
nowadays that uses this sort of these
sort of principles so ok so this is how
we model the light again we've got just
like before we think of the aperture as
being placed in the front focal plane of
the lens of course the lens is very
complicated it's got loads of pieces of
glass in but we we think of it just as
like a black box and we describe it just
by this surface here which is called the
equivalent refractive locus and what
that is is it's the locus of points
where this ray and this ray intersect
and and actually if you've got a well
designed system that is satisfies what's
called the sine condition this surface
will be a sphere but I'm not going to go
into the details of that I'm just going
to assume it's a sphere in in what we
come comes next so yet in this case
you've practically got lots of plane
waves incident in the focus but
unlike the paraxial case we really have
to take into account now the
polarization of these electric fields
and add together the electric fields in
the focal region and so this is what
happens after the lights gone through
the through the lens it starts off as
building being plane polarized so in my
diagram here the red represents the
electric field say so we start off with
the electric field being plane polarized
in the vertical direction and then after
it's gone through the lens the the Rays
a focused on the focal point which is at
the center of this sphere but if you if
you actually calculate what happens to
the electric field vector this is what
it does after you focus it the electric
field lies on the surface of this sphere
with this geometry that you see that
these lines all go through a point on
the far side of the sphere and the same
is true for the magnetic field as well
you can if you calculate what happens
this is what you find happens and you
can show that this is actually exactly
the same field as you would get if you
placed at the center an electric dipole
along this axis and a magnetic dipole
along the y axis so if you add together
those fields you would get this and oh
yeah and one more thing to say is you
can see that if you if you if you've got
a small aperture system lo na you're in
this region here and you can see now
it's becoming very close to being
linearly polarized again it's only when
you get to big angles that it starts
becoming a bit more complicated so
Richardson Wolfe in 1959 so this is the
same EMA Worf who wrote the book and
in their paper I haven't got the
referenced here I'm sorry but anyone
remember what it was published here
we've gotten no that doesn't matter it
proceedings of the physical society I
think maybe but anyway it doesn't matter
you can easily find it on the way with
you if you're after it they they they
showed you you found it okay Proceedings
of the Royal Society of London in 1959
they showed that the the field in the
focal region could be written in this
form so rather than just have one
diffraction integral we now end up with
three for the three for the defining
these three integrals and then these are
these are the electric fields in the
focal region written in terms of these
integrals i want i know i 1 i 2 and
these things are very similar to the
paraxial ray printer not normal
practical one is a bit similar to this
than the j0 and and this this is an
extra bit which comes about because of
the system satisfying the sign condition
and i'll draw it for you what's
happening basically if this is this at
Appomattox sphere surface if you put in
light plane polarized light into this
system you can see if this is uniformly
uniform intensity when it goes in you
can see that the amount of energy that's
gone into here and the amount of energy
that's going to go into this angle which
is very much bigger is going to be the
same so you see what I'm saying so this
amount of energy is going to be squeezed
into this angle whereas this this amount
of energy is going to be spread out over
this angle so so this is produces this
this factor here
and and then you can calculate what
happens so this is yes I'm sorry about
let you see you write things like that
not circularly symmetric but of course
what I mean is that that this is not
circularly symmetric this really should
be a circle before the modern technology
distorted it and and so rather than the
normal area disk what you get is this
elliptical focal spot like this and this
is another calculation these are both
from a couple of my papers this one is a
really old paper 1977 really old and
this one was from a bit more recent 1997
so this was written with piece of Tarak
when we came up here with a new way of
calculating it where basically we look
at the modal expansion of the field over
the sphere and express it as a summer of
these spherical harmonics the modes of
the sphere and so this becomes a very
actually a quite computationally
efficient way of calculating things and
this is for two systems then this is for
an angle of pi by three so 60 degrees
and this is for 90 degrees so this is if
you if you're actually the limiting case
when you're actually focusing a whole
hemisphere of light onto the focus and
you can see that the the sort of general
shape stays much the same but it changes
a bit as you change the aperture all
right the other thing I was going to say
a bit about is Bessel beams I don't
don't think George did it did anything
about vessel beings in the end but I did
see it as a title on some on the the
contents of some of the lectures as
though it was just about to come but
maybe I just missed it but anyway
this is a concept that is quite
interesting at the moment
basically there are three what three
ways I'm showing here of doing it one is
what happens if you put in your that
this is the aperture stop of this lens
and we put this sensual obstruction so
that it becomes just an annular region
and we make the width of this annulus
very very narrow then remember we just
in the last lecture we looked at what
happens for this annular lens and we
knew we know that what what happens is
compared with the airy disc the central
lobe gets narrower but the side lobes
get bigger well it turns out that in the
limiting case as this becomes very
narrow you get the the the airy disc
becomes given by this very simple
expression just j0 squared where j0 is a
vessel function so that's why it's
called a Bessel beam and and it has this
very nice property that actually this is
this satisfies the wave equation so it's
like a sort of mode of free space so
this Bessel beam propagates with an
infinite depth of focus it just
propagates without spreading and so
that's why these beams have become of
quite a lot of interest actually that
the concept is really quite old there
was a device proposed by MacLeod in 1954
for producing something rather simple
similar which is called the oxy con an
ACCI con is like a a conical piece of
glass it's like a conical prism and you
can see then that a prism reflects the
light
parallel like this and so you can see
wherever you are along this axis you can
see that what you'll see is like this
coming at the same angle and so you can
see from this how you get this
that's the increased depth of focus of
this system and another system was
proposed again a long time ago by Dyson
to do this not with a prism but with a
diffractive optical structure so you
make a sort of grating which is a bit
like a zone plate except that a zone
plate the Rings on his own plate go as R
squared whereas sorry as the width of
them the spacing goes as 1 over R
squared whereas this one the the Rings
are linearly spaced and in this case the
because the the grating is linearly
spaced all these rays go at the same
angle and you'll get the same effects as
this so so that's how you make Bessel
beams so I've been interested in in them
for a very long time and this is from
another of paper of mine going back to
1978 very little very long time ago and
this this here was a one of the lines of
that paper as it says a wave with zero
order bessel function radial
distribution propagates without change
and so what this is showing is what
happens if you instead of thinking of
the the paraxial theory what happens to
this vessel beam if it's not a paraxial
case if you put say this annular mask in
front of a high numerical aperture
objective lens and this is what you get
so this is for different angles 30 60 90
as before I've even gone further here
I've gone 220 and you might wonder what
that can possibly mean except it means
you can see that you could do it by
using a mirror the parrot parabolic
mirror that George Spota bats spoke
about earlier so you could have you can
have light coming in this way and say a
narrow annular beam coming in here so
this might correspond sides of this case
and then 120 case would be when the
lights coming in like this so you can
actually get a bigger than 90 degrees by
using a mirror sort of structure but
anyway what you see that happens if the
angle of the if the angle of the
focusing is small then it's pretty
similar to what I showed earlier for the
for the full lens full aperture but as
you make this angle bigger and bigger
eventually very nasty things start
happening this thing starts splitting
into two spots rather than one and you
can see that it does more and more of
that as you make it even bigger and so
this is a this is all assuming that
we're putting a plane polarized light
into the lens so what we the conclusion
we come to is that putting plane
polarized light into that lens is not
really a good idea not good news and so
this is explaining things a bit more
first of all look at the the black
curves
so these correspondent so a normal airy
disc is this one and the this other one
here corresponds to the narrow annulus
so you can see this is the Bessel beam
you can see it's sharpened up quite
considerably but with some rather larger
side lobes but then if you go to the the
high n a case so this is calculating it
for a numerical aperture of 1.4 so for
an oil immersion objective and you'll
find that for the annulus now it's
actually no narrower than for the full
lens so this affects the polarization
effects are really basically getting rid
of any advantage in this sharpening of
the spot that we get in this case but
there it turns out there is a way of
getting round that and that is to not
use plane polarized light going in and
what we use instead is putting right
radially polarized light we've if we put
in a radially polarized light this
corresponds to this magenta color so as
it says here radial I've also called it
here TM zero because it corresponds to a
transverse magnetic mode I'll show you
how that comes about in a minute and now
you can see for that case we're back
again to getting something very close to
what you would get for the practical
case so there's been a number of papers
which have exploited using radially
polarized input into your lens in order
to get a smaller focal spot so yeah this
is how it comes about why it's called
transverse magnetic so this is this is
radially polarized light sorry yeah the
red is the electric field so that's in
the radial direction if we add a small
na it would just be still radially
polarized but you can see that if you
look on the complete sphere it's that's
becoming like a light lines of longitude
latitude on the sphere and you can see
the magnetic field or these lines which
are purely transverse so that's why we
call it transverse magnetic and the zero
corresponds to the fact it's the lowest
order mode that's like that there's
another case which is azimuthal
polarization this is where the electric
field goes like this and the magnetic
field is like this so this corresponds
to te but I'm not going to say anything
more about that because that doesn't
produce anything useful and yeah this is
actually I said there are three ways of
making these Bessel beams there's
actually a fourth way which
some people are together with me at the
data storage Institute here in Singapore
have been working on and this is to use
a lens with some sort of binary optic
structure and actually this was from a
paper where we were simulating what
happens if you put radially polarized
light into this other system and use and
you optimize this binary mask in order
to get close to a Bessel V and these are
the sort of results you get this is
along the axis you can see that you get
something which really propagates
without spreading over quite a long
distance right so you know this is
showing how there are various ways of
specifying how the performance of a
focusing system and one way is is to
calculate or measure the electric field
at the focal point for a given amount of
power going in and what you want of
course is for the power of the focus to
be the electric field at the focus to be
as big as possible and so this ratio is
this thing called F here and what this
is showing is that this one here is
actually for our our plane polarized
light going in and showing what happens
to this ratio as you increase the
aperture of the lens and so you can see
that as you increase the aperture of the
lens so of course it focuses better and
better and therefore the same amount of
power getting going in you get a higher
field at the focus so that's why it
rises like this but it you see only gets
to eventually the maximum it ever gets
to is a half and the reason it gets to a
half is because I showed that this
polarization is actually equivalent so
an electric dipole plus a magnetic
dipole and so only half of the power is
going into the electric dipole
and then and it's the electric dipole
that gives an electric field at the
centre the magnetic dipole gives a
magnetic field at the center which we
don't want so only half of the power is
going into the right thing so that's why
that does that and then you can see
there are another two curves here this
one is our radially polarized light so
you can see that it's very slow to get
started here and that's because with the
radial polarize distribution you've got
very little power near the axis anyway
so but eventually it starts taking off
and you see that some angle here which
is really quite big so this is 90
degrees sent here somewhere so this is
going to be maybe 80 degrees or
something like that it overtakes the
plane polarized one and becomes better
but you can see there's another one
another curve here which is for just an
electric dipole on its own so a
transverse electric dipole so rather
than try and give this field we just
might try to get this polarization and
if we do that then we see we do even
better so in this region here you can
see this is doing better than either of
these other two so this is what we call
an electric dipole wave and this is what
this looks like they this mixed dipole
which is the electric plus a magnetic
dipole is equal to an electric dipole
plus a magnetic dipole so this is the
electric dipole field so what we need to
do is to make it so that after we focus
the light the electric field of the
focus light is on the surface along
these lines like this basically you see
that if you look in the region near the
axis all these behave the same they're
all going to be virtually linearly
polarized it's only when you get so big
apertures that these become different
and so this is now looking up what
happens with Bessel beams
so this is the this is the the the soap
what the so-called mixed dipole which is
the plane polarization case for 30
degrees 60 degrees 90 degrees 150
degrees this is for the electric dipole
case 30 degrees 60 degrees now you see
here this is this is a bit better than
this one and this one that this spot has
divided into two but this one hasn't
divided into two but it has got these
rather large side those developed here
and and then if you look at the this
other curve that these other ones here
this is what I called a te one so this
is another mode I haven't mentioned
before it's another te mode but it's the
TE one rather than the T 0 and you can
see this one actually is the same in all
these diagrams it's independent of the
angle there so you can see actually this
one turns out to be slightly smaller
than this one even and this shows what
the TE one looks like this is te one so
again the electric field is transverse
but it's not going all the way around
like te 0 it's going this way on this
side and this way on this side so if you
could make this polarization this would
produce that results I just described
there so this shows all these different
types so so what do you have to put into
the lens to get that so this is what you
have to put into the lens so if you put
in plane polarized light you get this
what I call the mixed dipole case in
order to get electric dipole polar eye
polarization out you have to put in
light that looks like this for the
magnetic dipole it has to look like this
see here it's going it's pointing in
here is pointing out and for the TE one
it's it's like azimuthal
and for the tm-1 it's radio like this
right so by altering the polarization of
the light that goes into the lens you
can you can generate all these different
mode patterns and this is what it does
this is calculating the area of the
focal spot as a function of angle for
all these different cases so here
there's a whole range of them for a full
lens full circular aperture and this is
for a few different cases for the for
the Bessel beam and you can see what
happens is we want this area of course
to be as small as possible and you can
see that actually in this region here
the electric dipole one is the smallest
and then the TE one is the smallest and
then eventually the radially one the
radial one becomes the smallest but if
this doesn't become bigger and until you
get over a numerical aperture of point
eight nine and then this is for the full
circular case I've got lots of different
cases here right this one corresponds to
a paraboloid mirror this one corresponds
to basically a different appetizer of
the light which you can see it actually
is doing better here than the others but
eventually runs out of steam when you
get to big angles so and you can see
here the the TE case is the smallest and
then eventually the radial case is the
smallest after you get above a numerical
aperture of 0.9 one right now the other
thing you'll notice though about all
those spots I was showing is they're all
not symmetrical so what can we do to
make them symmetrical
well this fits in a bit with what we
were talking about before the
polarization case the circular polarized
the T te and the TM 0 are already
circularly circularly symmetric
rotationally symmetric this one
corresponds to our normal radial
polarization but these other ones we
know that X polarized plus
why polarize equals circular polarized
so we can do similar sorts of things
with these teas or the electric dipole
polarizations we can add we can add one
of them along the X with one of them
along the Y with an eye there and that
actually is now going to give something
which is rotationally symmetric and
actually if you think a lot more about
it it corresponds so azimuthal
polarization with a phase singularity
with a vortex a phase vortex
superimposed on it and this one
corresponds to some elliptical
polarization with a phase singularity so
what do i mean by this
this one azimuthal polarization with a
facing clarity what I mean by that is
the and anywhere around here the the
it's polarized in the azimuthal
direction but as you go around here the
phase changes from 0 writes around to 2
pi plus no no no ok sorry it changes as
you go around here it changes from 0 to
2pi so where it gets back to here it
matches up again but here it's actually
changed by pi so it's effectively in the
opposite direction right so that's what
I mean by that there's a continuous
phase variation as you go around there
and and this one elliptical polarization
with a face singularity what it means is
that the light is anywhere on here it's
like elliptically polarized like this
say but at any any angle it will be a
similar sort of ellipse but the phase
change the phase of this ellipse will
change with a face singularity as you go
around in the circle and
yeah so what happens if you do that well
this is what you get so this is looking
at the cross-section for three different
angles for all these different types and
so that this is for Bessel beams again
because they're the simplest to
calculate so what we find is that for
small angles the radially polarized is
no good it turns out that the reason why
radial polarization works actually is
because sorry I should have said this
earlier but I don't know why I didn't
really why it works is because the light
that comes down here has got some
polarization in this direction here
which means it's got a component in this
longitudinal direction so so actually
the radially polarized light put into
the lens produces a longitudinal
electric field in the focal region and
that might seem very strange to people
first of all you can actually generate a
wave that's moving along which which has
got a longitudinal electric field but
that seems to be what is true and but
the point is of course if this angle is
small then this longitudinal component
is going to be very weak and that is why
this radially polarized one here doesn't
work well because the transverse field
is dominating over the longitudinal
field but as you increase the aperture
you see it gets better like that and and
then I also show here the the electric
dipole all of these are all the same on
this for this small angle but they for
the larger angles they become different
and and this is showing how sorry this
angle this that this arrow has moved
itself for some reason other know why
it's moved itself but this was supposed
to be pointing to this blue line here
and showing how in this case the TE one
case is that is the narrowest it's
actually even narrower than the radio in
this case here
with some rather slightly some bigger
side lobes here and this shows also the
electric dipole case as well and and
this one showing how the plane polarised
case is really very bad as we showed
earlier once you get some really big
angles and then this is comparing them a
bit more this is actually plotting the
normalized width of that rotationally
symmetric spot that you get again for
all these different cases and and
showing so these are the cases that are
full aperture these are the cases of a
Bessel beam and you can see that
actually it turns out that the TE
annulus is the narrowest the TE Bessel
beam and all the time for any angle it's
always narrower than the radially
polarized one and for these cases here
for the full circular case the TE 1 is
always narrower than the green one until
you get to this point here which
corresponds to a numerical aperture of
0.98 which is higher than you would ever
get to in any practical system so what
this is all showing is that as I say
putting plane polarized light in is not
the optimum it turns out they're very
often the optimum is can be different
according to the circumstances but it
can be this electric dipole polarization
or it can be the radially polarized
input or it can be the transverse
electric and I think I yeah this is this
is showing the strength of the side
lobes and showing how you know the the
electric the transverse electric might
be narrower but it might have bigger
side lobes
so here the Rayleigh repolarized those
smaller side lobes week aside those than
the that's the te right so and there's
our conclusions about this there so if
you focus focus plane polarized
you'll get a big focal spot so you can
improve that by using readily polarized
light and that comes about because
you've got this strong longitudinal
field on the axis but we find that
electric dipole polarization actually
gives a higher electric energy density
of the focus as I showed earlier bit
higher than you get with radial
polarized light and then I showed that
the TE the TE 1 mode polarization gives
the smallest central low and it's
smaller even than you get with radially
polarized light and but the TE on its
own is a symmetric but you can make a
symmetric version of it which is
equivalent so has a mouthfull
polarization with a face singularity so
that's about it for what I was going to
say about this so and so this is some
finishing up the course with something
which is really sort of ongoing research
and something that I think there's a lot
of people around the world working on
these different problems and trying to
optimize these systems to get the best
possible focusing properties that you
can get with a high in a lens and you
know I mentioned before all the many
different applications that this could
be useful ok so I'm going to stop there
questions any questions about this any
questions about earlier there's lots of
questions about this are there yeah
maybe like there's a lot of information
there I think you're probably all
suffering from overload but but it's
it's all very complicated actually but
but but very beautiful as well I think
it's it's it's it's nice physics I think