Optics (Lec - 25) - Electronics & Communication Engineering Video Lecture - Electronics and Communication Engineering (ECE)

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right okay so I think this was where we
were up to I wasn't here last time so I
don't really know but George told me
that this this was where you're up to so
that's where I'm going to carry on from
and so this section is all about imaging
and resolution and it starts off here
looking in some strange American
Dictionary the word the definition of
resolution of course you know myself
coming from Oxford I would always use
the Oxford Dictionary I don't know what
that says but anyway this is quite
interesting the the verb here resolved
to break up into constituent parts to
analyze to find an answer to - to solve
determined decide on but anyway I think
this first part is the very first thing
is the is is a good one isn't it so
break up in it into its constituent
parts so you look at an image and you've
got to actually determine what the
different parts of the image really mean
and if they sort of merged together you
won't be able to resolve what the image
is trying to say okay so that's what it
means and yeah there are lots of
different ways of determining resolution
different definitions that people apply
the most used one perhaps is the
Rayleigh resolution limit which was
proposed by Lord Rayleigh Lord Rayleigh
was very interested in astronomy and so
he was interested in trying to quantify
the imaging of of stars so if you look
at up a telescope at the sky and you've
got two stars very close together then
you'll see like the two images of the
stars this shows two different
separations of those two points
and now it's important to note that
stars of course two independent stars
will be completely incoherent with
respect to each other because they
obviously don't know the one doesn't
know what the others doing right so they
emit light incoherently with respect to
each other so to calculate the image of
these two points you have to add
together the intensities of the two
images and that's what this shows here
and so this shows two different spacings
in this one here this is what I think is
really the normal normally taken as the
Rayleigh resolution limit I don't know
about this other one I don't know where
George just got this from but he talks
about these two different limits in
these notes so in this one here you'll
see that this peak here is placed
exactly over the zero of the other of
the first zero of the other image right
so that's what defines this separation
shown in this picture and we find that
if they're separated by that amount the
total image is given by this blue line
we get that just by adding together the
two intensities of the two points
together right and you can see that this
drops a bit at the center what is it
it's I think the intensity of there is
point eight eight eight
except I can't remember there no it
doesn't matter
but anyway it's about point eight isn't
it a point eight three five that's that
so that finger serves but anyway the
other interesting thing of course is
that because this one is zero at this
point it means that this maximum is one
it's exactly one it's it's a it's just
the one point we know that no
contribution from the other one right
anyway so that
that is arbitrarily taken as being the
definition of the points being just
resolved so we say that if they're any
closer together than that you can't
resolve them you would see a little bit
of a dip still but we we say that the
dips not really big enough for you to
really be confident that there are two
objects there and if you make them
further apart then of course they'll
become more distinguished and this shows
an example here where you can see now
that this is twice the separation so so
this is where the PSF diameter equals
the points or spacing right so so they
what does he mean by that
sorry the hit this one the PSF radius
equals the point source spacing yes okay
so that is the distance between the
center of the image and the first dark
ring in this one the PSF diameter equals
the point source spacing no it's not the
second zero it's not at the second zero
exactly anymore is it okay so the
spacing here is twice the spacing there
anyway I think that's what I'm taking it
at this meaning but as shall in just
pointed out that means that this is not
actually at the zero this is not placed
at a zero of this other one because the
if you remember the zeros of the airy
disc are irregularly spaced aren't they
right so they're not equally spaced like
the light for a sink and so because this
isn't zero at this point here this
maximum here will be not exactly one it
will be slightly more than one okay
and if so so if this condition is
satisfied
then in this case that the Delta are the
the where does he view find that that
that's the I guess it's this distance
here isn't it it's the separation of the
piece yeah okay it's the separation of
the piece is point six one lambda over
na oh yeah okay so here is the are are -
they're obviously coordinates in the
object plane and the image plane
effectively yes so of course it depends
on the the the numerical aperture in
those two spaces won't be the same of
course but there will be magnification
as well which is gonna mean that you get
the same answer for the resolution in
both planes so this this point six one
as this magic figure that comes about
basically because of the it's related to
the first zero of the Bessel function J
1 and the first zero of the Bessel
function J 1 is three point eight three
and this is three point eight three
divided by two pi that does that sound
right
nothing yeah yeah I think that sounds
right this one here is twice the spacing
so one point two two lambda over na and
so this is three point eight three
divided by pi yeah so here actually
these are obviously extremely well well
resolved you wouldn't have any problem
in seeing that there's two points there
okay and so yeah so here here he calls
these two definitions the safe
definition and the pushy definition my
experience is that the pushy definition
is the one that is normally quoted in
places and so this is this one where you
can only just see a bit of a dip in
between them okay note that if we did
the same thing for the 1d system ie
for a pad share rather than
aperture then you'd be looking at the
first zero of a sink rather than a
Bessel function and so you get these
expressions rather than these so one
point so two rather than one or not 0.61
rather than null point five so in this
case you actually you can see you can
resolve slightly smaller spacing with
with a slit aperture than with a
circular aperture and and this is a true
effect that sometimes people try to take
advantage of okay you will see that
different authors give different
definitions right so yeah so I'm not
going to go into all that because it's
not really important at this stage
Rayleigh in his original paper noted
that the issue of noise and warned that
the definition of just resolvable points
is system or application dependent
actually what you find well a lot of
that is true of course here we're you
know it very much depends on the actual
system and what you're trying to measure
all these sorts of things and you can
come up with lots of different
resolution criteria for different shaped
objects and the comparison of different
systems doesn't always give the same
answers according to what you choose so
beware there's not one answer to this
question of resolution the other thing
is as he says here is noise is an
important thing of course if you add
noise to these image we've got some
examples later where there's noise added
to the the images so that you can see
what it does to it but one thing that is
perhaps not quite what George is saying
there actually you do find if you look
at the image of these two points as you
bring them together you actually do find
that the this this degradation in the
image is really rather sudden so
although okay it's not a specific point
it is it is
quite sudden and so you'll find that if
you change the separation around
slightly more or slightly less than that
you'd find that this since this central
bit would go up and down quite a lot
actually so it is quite the sensitive
change around that spacing so that gives
us some idea that that may be this this
resolution limit as its defined does it
doesn't mean something anyway right now
that's for a perfect aberration free
system but very often you've got systems
where you've got aberrations and that's
obviously going to mean that you're the
resolution is not going to be as good so
this picture shows an example with coma
so this is with an off-axis ray off axis
beam going through a lens and you get
this this coma spot here so now of
course this spot is bigger than it would
be if there were no aberrations and
therefore you'd expect that the
resolution will be worse so this is
another thing you have to take into
account I don't think there's going to
be much more about that what did that do
something then or didn't push it fast
enough right now the other the other way
of looking at the effects resolution and
the effects of aberrations is looking at
the transfer function approach and so
this is this is giving results for a
one-dimensional system and it's it you
see it's an MTF modulation transfer
function so that means it's an
incoherent system right the the MTF is
the Fourier transform of the point
spread function right so this is for the
aberration free case this is a sinc
squared the point spread function for
the 1d case and if you take the Fourier
transform you get the
the modulation transfer function as he
calls it and vice versa of course these
are Fourier transform pairs this is just
a triangle function and and this shows
what would happen if you've got some
aberrations there in general what would
happen is that this would be broader and
this would be deformed so that you've
got a lower spatial frequency response
here actually he seems to be showing
something which has got an improved
response at the very high frequencies
which is quite interesting because you
know what that would mean actually is
that if your objects only had those high
spatial frequencies this the aberration
the system with aberrations would
actually give a better image than the
one without aberrations so maybe that's
a bit anomalous okay alright so now some
common misinterpretations attempting to
resolve object features smaller than the
resolution limit ie
1.2 to lambda over na is hopeless so
he's saying that that's not strictly
true because this is not for a start is
not a sudden cutoff and and it depends
on the presence of noise and other
features so there's a very big no to say
that yes you can sometimes see things
which are smaller than the resolution
limit and and then also another point is
that there are ways you can actually
improve the performance using for
example the clean algorithm the clean
algorithm is a thing they use in
astronomy for doing the convolution it
works particularly well it will be very
good actually for some of these
microscope things that people do
nowadays with these very sparse space
molecules because it's designed to work
with you know in astronomy with isolated
point images all right so if you looks
at a comply
continuous objects like a motor car or
something like that it would wouldn't
work it has to only work on on point
type images then Ravena filtering will
say something more about that
expectation expectation maximization etc
so there are various digital ways you
can improve the image and get
information out of the image even if you
can't see it by how are I and then
there's this other words super
resolution super resolution is it means
basically getting better resolution than
you're supposed so I beating the
Rayleigh diffraction limit and various
ways have been proposed for doing this
and until recently I guess most of them
were pretty unsuccessful but more
recently nowadays some of these
techniques are doing really well on I
think the most noticeable one probably
is stead stimulated emission depletion
microscopy where where Stefan Hale is
getting resolutions which I don't know
if more than a factor of 10 better than
the new shirred according to the
Rayleigh diffraction limit but anyway
here we hear he's talking about what are
called super super resolving filters
what you do is you you alter the pupil
function of the lens by some clever
trick in order to sharpen up the point
spread function and I think that maybe
George did mention that earlier on at
some point but the problem is what
always happens is that it doesn't
actually you know and you never get
anything for nothing
that's the problem you make the set that
you make the central lobe this essential
low snare OA but you find that the side
lobes become stronger and the other
thing is that very often the power
transmitted through the system is much
reduced so you can
imagine if you've got some filter that
absorbs part of the part of the power
then you you're wasting power right so
both of these can be bad and so
consequently this idea of super
resolving filters in practice is never
really up until now been very successful
from a practical point of view but
anyway this shows one example this shows
us a circular pupil and and there yeah
so and then this one is is is a an
annular pupil which you can model as
being one circuit circular function
minus another one and of course once you
write it in that simple way as a circ
function it as the difference between
two search functions you can do the
Fourier transform of that very easily
and get the point spread function of
these two cases very easily and so he's
working it out for a focal length of 20
centimeters and a wavelength of 0.5
microns and the width of that aperture
is what does it looks about it looks
about four millimeters diameter
something like that yeah anyway so this
is he worked out what it looks like so
this is the PSF this is supposed to be
an airy disc it's been sampled a bit
coarsely so it looks a bit ragged and so
you what you see is when you when you
look at the annular case you put this
central central obscuration in the
center there what it does is it narrows
up that central lobe and and you can see
though it also increases the strength of
these side lobes that these are much
higher than these ones so although in
some cases this can give a good
an improved image compared with this for
example for point objects it would do
well but if you've got an extended
object like an image of a car or
something it might not work so well the
other way of thinking on it into is in
terms of the MTF and this is our normal
MTF for a circular pupil it's this
so-called Chinese hat function and then
for the annular pupil this is what you
get you remember that you you calculate
these this is the convolution the
autocorrelation of a circle right so you
have two you get two circles and you
displace it one relative to the other
you calculate the area of overlap and
you plot the area of overlap as a
function of the distance between the two
centers so that's how you actually
calculate this and you do the same with
this one and you can see what you get is
you see that it drops more quickly first
of all this is normalized to one because
you normally normalize transfer
functions to one for 0 spatial
frequencies that's the way we normally
do it but I guess maybe maybe it's not
the only way of doing it this drops down
quickly because you can see if you're
working out the autocorrelation of this
annular aperture you can see that you
mean that you move it a small amount the
the one the one bright region is not
going to overlap with the other one
anymore right so that's why it drops off
quickly there and then it has another
blip out here which basically comes
about from when the two stat the two
annually where they've both got their
bright bits on top of each other right
and so if you compare this one with this
one you can see that here the spatial
frequency response is much lower here
actually it can be higher the cutoff of
course is still the same right
so you could calculate the image of any
object you fancied from from knowing
that and this is another example this is
the case of a Gaussian apodization so
what we're now doing is instead of you
know the the the annulus is effectively
boosting up the edges of the pupil which
is why you get this improved high
spatial frequency response here we're
doing the opposite of that apodization
if George was here it would he would
tell us
in fact he already did in one of the
previous lectures apodization means to
cut off the fate in Greek I believe
he'll probably be listening to my
lecture and tell me I've got my Greek
all wrong but anyway it's to do it it
means to cut off the feet and what it
means that why it's giving that name is
because it cuts off the side lobes so
you see the side lobes of the airy disc
disappeared now so by making this shaded
aperture in this case it's a Gaussian
function you make something which is a
bit broader actually in this case
doesn't look very much broader at all
but the side lobes have gone and and
this is what it's done to the to the
transfer function you see now it's it's
slower to fall off here than this one
but the high spatial frequencies are now
reduced relative to this other one okay
so overview on this then this is what we
call or what George is calling pupil
engineering and so so if you try to make
the main the central lobe of the PSF
smaller then you're going to get bigger
side lobes and that seems to be
something that you can't avoid nobody's
come up with a way of doing something
clever that avoids that and the opposite
of course if you make the main side
lobes main side lobe bigger the side
lobes get smaller
no doubt that is maybe not so
necessarily going to happen I suspect
that if you really wanted you could
probably make something that had a big
central lobe and big side lobes but
nobody would really want that anyway ok
and and then also this power loss
problem that there is going to be some
worse signal-to-noise sort of
performance right so angular and euler
type pupils typically narrow the main
lobe at the point spread function at the
expense of higher side lobes Gaussian
type pupil functions typically suppress
the side lobes but broaden the main low
and so as it says here different sorts
of filters like this might possibly be
worth considering it for certain
particular applications yeah and then he
gives another couple of problems with
these things making a filter which is a
varying amplitude is not easy and I
guess nowadays you might do it with some
liquid crystal device hopefully we might
be doing that so when we shall in and
but but you know and until recently this
would have to have been done by some
sort of photolithographic process and
actually getting controlling the
absorption of the filter to accurately
fit what you designed is not trivial and
there's also this energy loss problem
zone yeah nowadays people are very
interested in in phase filters you can
do a lot of these clever things also by
using phase masks rather than amplitude
masks and of course a phase appetizer is
going to be lossless so that might make
it sound attractive and so George says
that maybe it maybe it is attractive
actually it turns out I think it's not
quite as attractive as it might sound
because actually very often what happens
is if you use an absorbing filter the
absorption that you can introduce
actually can reduce the sizes the size
of the side lobes so if you do it
cleverly you might be able to make the
side lobes
lower and also still get as much energy
into the central lobe with it with an
absorbing mask that's with a pure face
mask right yeah and then we get on until
these terminologies that George is
obviously annoyed by as much as me this
is one this super cool digital camera
has a resolution of 8 megapixels and so
what's he gonna say about this big now
again right so this this is using the
word wrongly it's this this is nothing
to do with resolution and so what they
are actually referring to is the space
bandwidth product of the of the camera
how many pixels there are which is not
really measured as resolution the other
the other one as all the people in my
group would know that I always get very
cross about these these interferometry
people who are always talking about
interferometers being able to resolve
one angstrom which is also wrong because
it's nothing about ret resolving all
rights so resolving at resolution we've
said what resolution made okay so this
question of the number of pixels is
there a connection between the toe well
I guess the answer is there isn't there
isn't there and there is a relationship
if you design the optical system
properly but what what some George is
pointing out here is that actually you
know that this is an example where the
pixels are very much smaller than the
point spread function of the
of the optical system so actually here
you're very much over sampling this
image so all you're measuring you're
measuring very accurately a blurred
picture so that's not really giving you
very good resolution they're not really
very you know the the number of
resolution image resolution elements in
your image is actually much smaller than
than the number of pixels sometimes they
use this word rez else don't they
resolution elements have you seen that
they spell it re SEL it's an
abbreviation for resolution element so
it's the number of resolved spots you've
got in your in your device rather than
the number of pixels okay so more
misstatements it is pointless to attempt
a result beyond the Rayleigh criterion
however defined no the difficulty
increases gradually as feature sizes
shrink and difficulty is noise dependent
so it's not a sharp hard and fast line
but that said I think you I think it
probably would be pointless to attempt
to resolve 10 times the radius with an
ordinary optical system because you know
you're not going to be able to do that
but stead that we were talking about
earlier does do that right so this is a
neat a twirl okay so maybe it is worth
attempting to do it because people have
come up up with ways of doing it now
Abid ization can be used to beat the
resolution limit imposed by the
numerical aperture and there's a big no
there again watch the side lobe lights
and the side lobe growth and poor
efficiency loss I don't know if you
noticed but I suppose us what one and
maybe shall in and I can't remember now
there was a paper in Nature Methods
recently by Nikolai's AG live where he
gave me an example
of a super resolving mask and he gave
they picked this picture of a very
narrow point spread function you know
improving on the Rayleigh rate
resolution limit by a factor of I don't
know a few I think but I did so I did
some calculations on the design he gave
it turned out that this this little
central peak was surrounded by side
lobes that went up to something like 10
to the power 10 to the power 40
something or something really ridiculous
and and the amount of energy that went
into this central spot was like nothing
and what made it even worse was that if
you actually calculated what happened as
you went out of focus you found that you
also got these big walls around it very
high all around it actually so it was
this like this very small bright spot
completely surrounded by something which
was like 10 to the 40 times higher so
how you would ever use that in practice
you can you can think it really
obviously wouldn't be a very practical
device right and then right then the the
number of present
excells in your camera is not the
resolution so what is resolution okay
our ability to resolve to point objects
based on the image however this may be
difficult to quantify resolution is
related to the N III is proportional to
the NI the distance is inversely
proportional to the NA that's another
thing that people sometimes get into
knots about of course because resolution
means that the bigger the number the
more is the resolution but actually the
smaller the size the more the resolution
so beware so sometimes actually people
use the word that the term resolving
power
to get round that problem because
otherwise people say things which they
don't really mean sometimes yeah so okay
so other factors that can affect
resolution aberrations apodization and
noise right so is there an easy answer
when in doubt quote no point six one
lambda over an a and then you get the
marks in the exam okay so that's that
one done so how are we going got bit
more time yet but that was last week's
lecture I just gave there in anyway okay
so I don't know how far we're gonna get
with this one but we can at least start
it okay so so what we're supposed to be
doing today more applications of the
transfer function a bit about depth of
focus and a bit about the convolution
and there's some nice not some nice
simulations that George is done to show
that and then he's got down for
Wednesday which I'm also going to be
giving I think George is not back do you
know when George comes back no no anyway
so I shall be giving the lecture on
Wednesday and so a polarization what we
don't finish from today will obviously
also be on Wednesday and then there's
polarization intensity distribution at
near the focus of hai na imaging systems
and yeah I don't know what else we're
gonna do actually because George keeps
changing his mind about what we ought to
put in that last lecture okay so D focus
so this is an example of D focus anyone
know what this movie is I don't I'm I'm
out
out of touch with the cinema recognize
who they owe the stars are no no I'm
only asking because I just did this fab
Club okay right anyway what you'll
notice is this guy here is it a guy I
can't really recognize what it is
actually but it's out of focus so yeah
it could be yeah okay so anyway so
that's an example of D focus right then
this is what happens when you get a lens
in the paraxial approximation and look
at the intensity in the focal region you
get something that looks a bit like that
it doesn't look quite the same as born
and walk of it I don't know how George
did this calculation but but you'll
notice that this is the focal plane so
this is a cross-section through the airy
disc so you see the central ah well I
can tell you what why it's different
then you can see these have got
regularly spaced zeros so he's done this
for a slit aperture as he and so that
might explain why it looks a bit
different but what you see excuse me is
that as you go out of focus well note
one thing it's symmetrical about the
focal plane and and this is I think we
probably had that somewhere before
actually that it's symmetrical if if
it's the if it's calculated using the
usual Debye approximation you you get a
result like that so if you've got for
example the aperture stop is in the
front focal plane of the lens and you
look at what happens in the back focal
plane then that would be the case you
get this symmetrical behind me like this
and you can see also these sort of
bright back
that seemed to go off diagonal lines
here I guess yeah so that corresponds
roughly to the edge of the aperture then
of course right and oh yeah so the width
of the central spot in this direction
then we've said is this naught point six
one lambda over na and this distance
from the focal plane where you get to
this first zero is given by some
expression like this it goes as if this
is really only true for the practical
case but for the practical case this na
here na squared here if it's the high
aperture case which we might we might
consider on Wednesday you'll find that
this is not quite right anymore okay
yeah so Delta X is the radio resolution
Delta Z is the depth of focus today's
topic topics depth of focus or depth of
field and both of those he calls DOF
note of very high numerical apertures
the scalar approximation is no longer
good the vectorial nature of the
electromagnetic field becomes important
that is true there's there's other
things as well though when you do all
this practical stuff if you remember
you're all the time you're replacing
sine theta by theta and things like that
making approximations like that so there
are actually lots of approximations you
make what one is the the fact that the
angles are small the other is that you
and neglect the polarization effects
when the apertures become big then the
polarization effects become very
important okay so we're going to look at
this sort of system and so this is a
four f system with magnification and so
the two focal lengths of the two lenses
are different in this case and so this
shows what happens if you if you're
looking at
an object's which is placed a distance F
in front of this lens and your screen is
placed the difference f2 behind this
lens and your aperture is f1 from this
one and their two from this one so
everything there is set up properly
right for as as defined for a four F
system and this is the case where you'd
expect as I was just describing the the
D focus point spread function is going
to be symmetric all right so we place
our mask there and we've got our pupil
so this is our object function this is
the angular spectrum the the spectrum of
the object in this plane plane here
which is then filtered by the aperture
stop and then that gives some some sort
of amplitude in this plane here which is
going to be inverted and also magnified
or otherwise according to the ratio of
f1 over f2 and there's the Rays going
through it and this is showing how yeah
this plane this is his terminology this
is X this is X Prime
this is X double Prime and you can see
here X double Prime maximum is equal to
the and the radius of the aperture so
this is his terminology we're going to
get that in a minute that coming out
okay this is this is a function of X
this is a function of X double prime
this is a function of X prime this is
the numerical aperture of the lens
that's looking at it and yeah so okay so
the numerical aperture is equal to X
double double prime maximum which is
equal to a over f1
again assuming that sine theta or tan
theta at both visa right now what we're
going to look at now is what happens if
we do focus this system so now our
objects has been displaced the distance
Delta away and we want to calculate what
the image of that looks like now and so
what we can say is that after we've
illuminated this object we've got this
GT of X and then that light's going to
propagate from here to here and in order
to calculate that we have to convo vit
with the propagation kernel right so you
convolve it with this thing
it for some reason when George converted
this from his whatever he's other things
call keynotes to power point all the
equations like as though someone's had
their martinis before the lecture and
right so we then look at the Fourier
transform of that in this plane here
so this think involves the Fourier
transforms to this the convolution
transforms to a product and this
Gaussian transforms to another Gaussian
and notice that as usual of course you
remember the the scaling thing for
Fourier transforms the bigger a function
is the smaller its Fourier transform and
vice versa
so this lambda Delta at the bottom
appears now at the top right so the
bigger the bigger dowser is the bigger
this things going to be here yeah
we got yeah is e to the imaginary X
square a Gaussian or only the e to the
real X square gosh okay yeah the real
Gaussian is e to the minus x squared
isn't it
e to the minus x squared does that this
is this is what you might call a an
imaginary gas here but it all comes out
in the wash
actually so just the same as the Fourier
transform of the normal Gaussian is
another Gaussian so the Fourier
transform of this imaginary Gaussian is
another imaginary gas here right so
what's really converting is a parabolic
face front here to a parabolic face
front here yeah so it you can get it
just by suggest by getting the normal
expression I mean it's it sometimes
amazes me how much liberties you can
take me these things but you just take
the expressions for the Gaussian and you
put in this complex number and it gives
the right answer so it's very nice I
guess occasionally there might be
occasions when it doesn't work yeah okay
so what is then saying is that you can
think of this that this is the object
for our this is the spectrum for our D
focused objects which is multiplied by
the the pupil function of the system but
that is exactly equivalent of course to
thinking of it as the the ordinary
object spectrum multiplied by a a
modified pupil function right so it
doesn't matter whether you associate
this with this or with the pupil
function so so that's what we can think
of we can think of the the objects as
being the same as when if it is if it
wasn't D focused and we can think of the
pupil function of the thing being
modified to take into account of the D
focus we sometimes call that
the focus transfer function and so he's
now going to calculate that yes so the D
focus he calls it here that the focus
amplitude transfer function right so
remember here we're dealing this is
dealing now with coherent system we
before when we were doing the resolution
we were looking at incoherent systems so
now we've gone back to coherent systems
so you have to be wary of that because
sometimes there's some differences okay
so all this is saying they yeah oh you
remember to go from the from the pupil
to the transfer function all you have to
do is a coordinate scaling right so so
the the amplitude transfer function is
just the scaled version of this pupil
function and again it comes out to be
now of course it's going to be its
multiplied of course still by the
circular aperture but inside that
circular aperture where it's got some
value its value is given by this complex
exponential complex complex Gaussian
like we were just saying before and
that's what this looks like now notice
so this is cosine this is looking at
just the real part so actually it was a
complex it was an e to the minus I
something so you can break that up into
cos plus I sine and the COS part is the
real part and the sine part is the
imaginary part and as we when we looked
at imaging earlier the the real part
images real objects so an amplitude
object will be imaged by the real part a
phase object would be imaged by the
imaginary part but we're not going to
say anything about that we just keep
with the real part so this is cos but
it's not a normal cause its cause of an
x squared so that's why it looks a
rather strange shape compared with a
normal cosign
you can see it's become very flat there
and then it becomes more oscillating
later on and so this is what it looks
like
of course the actual scaling of this is
going to depend on on the the value of
this Delta right so here is looks at
where this crosses the axis so this this
parameter X double prime over lambda F 1
is equal to you the spatial frequency so
this is just u squared here and he says
that this crosses the axis where this
argument is equal to PI over 2 and if
you solve that you'll get this and so
that's where it crosses the axis and
then you have to multiply this of course
by the the this the circular aperture
right so this is the circular aperture
this is the region where the object
spectrum is non zero and so you can see
that only the parts of this blue curve
up to here are going to have any meaning
the bits out here are not going to do
anything so this is obviously looking at
a case where this is almost flat if you
made it a bit smaller even it would look
even flatter so this is a not a lot of
defocus and so here he's got then yes so
hit now is labeling where that cutoff is
so this is equal to na over lambda as we
know and in terms of the X prime
coordinate is given right by this thing
here so this is talking about what he
calls mild defocus and the condition for
that to occur is that this has got to be
smush smaller than this and then
yeah and that corresponds then in terms
of Delta so we've now got a condition
for this to a happen the value of Delta
has to be small compared with lambda
over 2n a squared right so you remember
this is what we said was the depth of
focus the depth of field or depth of
focus of the system depth of field is
measured in the object space ie doesn't
say anything about that the focus on
that one I thought I was he says a bit
about that later this is showing there
in another case then this is where the
much more deep focus now so now it's
oscillating a lot more wildly so now
this cutoff is much smaller than this
and so this is where you've got a lot of
the focus and the fact the fact that
this goes negative is probably the worst
thing it means that some spatial
frequencies are going to be image with
the wrong contrast so when you try doing
your Fourier synthesis to add up to make
me make the image it's not going to work
right and the other point that he makes
is that the regions around here of
course these spatial frequencies around
these zeros are not imaged at all
so that's another reason why it doesn't
give a very good image right so so this
is the case of strong D focus where
Delta's much is bigger than this thing
and this is already much bigger okay I
think we all just stop there we've we've
run over time or anyway so let's let's
stop at this point and we'll pick that
up next time