Optics (Lec - 23) - Electronics & Communication Engineering Video Lecture - Electronics and Communication Engineering (ECE)

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a good morning and today we're going to
start with a demo and we're going to
show experimentally some of the
principles we've learned about for
assistance in the past classes and
actually we're going to see a comparison
of an incoherent versus a coherent 4f
system when imaging resolution target so
but since now we have all the tools to
understand finally the setup here let me
just describe it I've been showing some
of these components over and over over
the past demos but I guess now we can
understand it so let me try to focus the
camera so let me start with a this is a
green laser so the first component that
we have here it's a ND filter that its
job thank you so this component its job
is to is to control the intensity of
light so we can attenuate it in this
case for excitement this is an
interesting component called a spatial
filter that is composed of a microscope
objective and a pink hole and now that
we know what for your transforms are the
microscope objective is essentially
taking the fully transform of the laser
beam which ideally is a plane wave so it
should be a delta function however since
this plane wave is not perfect has high
frequency noise so then the pee hole
what it does is does a low-pass spatial
filtering so basically removes all these
high frequency signal and after this we
have a nice spherical wave coming out
this component hues and iris control the
diameter of this aspherical wave and
this element here is my collimating lens
that will transform this spherical wave
into a plane wave in this case this is
just a beam splitters I will allow me to
couple from this side where I have a
wide light source in this white light
source we have a green filter so then we
only have green light coming out we have
an iris again to control the diameter
right now I'm blocking it so it's a beam
block and these are collimating lens so
here it should be the planer equivalent
of the incoherent side this is a planar
equivalent of the coherent side these
mirror here
bends the light 90 degrees to send it to
the side from here this is the one line
of the coherent line and then should go
into the common path of the coherent
line then they basically Bend again 90
degrees and this is the 4f system
section so here we have first an element
which is the object that we want to
image the object is called a resolution
target and it's a it's a clear substrate
so a piece of glass that has patterned
chrome lines of smaller and smaller
sizes that we're going to see then we
have our first lens of the 4f system
this is a non magnifying for existence
of these two lenses are exactly the same
focal length so first lens then this is
an aperture in the food airplane because
we're going to start with low pass
filtering the second lens and then we
are imaging these into a detector which
is this ccd CMOS array ok so now we're
going to switch to their camera and
start with a coherent case
so the first case it's like like what we
see in class we're going to illuminate
this object with an inline plane wave
okay let me just zoom in a little bit so
so this is the resolution target and is
an imaging condition right so the object
is in the front focal plane of the first
lens and the detector is in the back
focal plane of the second lens so what
you can see here is we are actually
imaging all the way two groups to 13
this resolution target has lines even
larger which I wanted to that we don't
see here and we the first thing that we
see the following we see some fringes
which are produced by the interference
from the glass so it's basically the
reflection of the plane wave interfering
with each other so this is very typical
in in a coherent imaging we see other
rings that are produced by by dust
particles maybe along the way so all
these fringes right now look destructive
to the image and perhaps they are for
certain applications however in for
those of you doing digital holography
you you notice that these fringes
contained a lot of information that is
encoded about the app or the amplitude
and the face of the signal that we can
exploit so just for comparison i'm going
to zoom in and see what is the maximum
that we can see here so you can see that
we can have a resolution other we know
what the optical resolution is we're
fairly with good accuracy we see maybe
down to the side of the group number 5
alright so in the homework we learn
about the low low pass filtering in for
your domain so used to remind you
essentially what we're going to do is
that the first lens is taking the
Fourier transform and it happens in the
meat in the food a plane so then in
there in that plane we put an aperture
centered and then what we do is that
we're going to start reducing the
diameter of the aperture to try to low
pass filter the signal more and more so
why do we expect
to see here well you did it in the pset
so we expect to see some blaring
progressive blaring so I'm going to do
it and hopefully you can see it from
there so right now it's all open and I'm
going to blur it so I'm going to do it
again this is clear blurry so I'm
controlling with this be Cyrus the and
you can you can see it's how the signal
that contains the higher spatial
frequency in this area goes first then
these one and this one but all of them
now if I actually this blurry condition
if I try to swim back again now we see
how the signal is really blurry if I
open the aperture becomes clear again so
now let's do the comparison of the exact
same thing what would you think it would
happen if i switch illumination out to
incoherent light so i'm going to block
the coherent light here and then the
first thing that we notice is that we
don't have those fringes anymore right
so it looks like a very clean image and
so again in this case you can think
about that for microscopy this is very
nice however from the digital
holographic point of view we don't have
that extra information that we can use
Indian coding of the signal now let me
zoom in to compare allow the resolution
in this case so you can see it's hard to
tell if we don't do a quantitative study
to see exactly where the what is the
smallest line that we can see in each
case but we can see the comparison again
versus the resolution in the coherent
and incoherent cases and now let's do
the low-pass filtering so again this
aperture why now is fully open I'm going
to started closing it and I'm closing it
closing at closing
open again close it so we see two things
remember that the incoherent system is
basically regulated by the OTF are
supposed to the 80s in the coherent
system so now that you can think of it
the DC term is it's it's it's a combo
because the ogs is the normalized cross
correlation of the of the atf which
means that when i close this aperture
you can see how the object the signal
becomes weaker and weaker because
essentially an restricting the DC
component that happens to be in all in
all the signal there but also gets
blurry it's hard to see but it's getting
blurry not it is getting blurry at a
different rate at the other system so
now let's go to high pass filter and I'm
going to switch back to the coherent
case so this is the coherent case so
right now what I'm going to do is I'm
going to remove the aperture in the food
a plane and I just did a very improvised
high-pass filter which basically is a
microscope slide that has a little piece
of tape in the center just to block the
DC component or kind of but anyways it
works
and let me somewhere around here let me
crank up the light and we can see how
basically we remove the DC term and you
see the high-pass filtering effect which
is essentially accentuating all the
edges of the signal now I'm going to put
the incoherent case and the coherent
superimposes to show edge detection so
this is a normal single and we can
accentuate the edges so what is
interesting also here is that if you
move the filter in different places you
can say just accent rate the edges in
the horizontal direction or in the
vertical direction etc etc we see that
they cohere incoherent case we don't see
the same effect which is as expected
again he's a normalized cross
correlation but with a coherent case we
do see this drastic change okay I think
Thank You papa any questions about the
demo or about coherence in coherence
yeah how do you tell whether something
is partially incoherent or like
completely incoherent okay so the test
is with one of these interferometers
that sabacc saw the last time you
basically measure the fringe contrast
for example if you're worried about
spatial coherence you set up a young
interferometer a mach-zehnder and the
fringe visibility will tell you if they
feel the degree of coherence it is full
contrast it is fully coherent and of
course you never see full contest that
you never receive food you know so it's
always somewhere in between that you
okay so so today we will have a
flashback of genetical optics that will
go back to something we saw in
dramatical optics and that's the single
lens imaging system Oh before I forget
and on Wednesday it is the project
presentations for the Graduate version
of the class so I wanted to remind you I
will also send an email from the website
but if you haven't done so already you
should have a meeting with your project
mentors finalize your presentations and
then on Wednesday what we'll do is we'll
be like conference tile each group will
have about 15 minutes approximately 12
for the presentation approximately 34
questions and white hah my groups I
think we have the six or seven group
yeah so I typically these things run
over so so will take probably fit full
two hours and with the presentations so
for those of you in the 271 you can have
to attend but I think
be very useful if you do because this
sort of interest in it is more advanced
material the graduate students will
present some topics from current
literature and you get an idea not only
of the basic optics but to do here but
also what kind of things happen in in
the research front for example last week
several of us pepper van and so on where
the conference in Vancouver so so some
of the topics that will be presented on
Wednesday were actually big big items in
the in the conference the conference was
actually called advances in imaging the
spring congress of the optical society
anyway ok so so going back to the
lecture this is something that we've
seen before but with geometrical optics
now we do with that understand how the
same system works but but with the wave
optics and if I can have the the tablet
again please so the wave analyze the
system of course is i think is we
basically have to to propagate using
final diffraction to propagate the
fields from one element to the next
until we reach the end of the system so
for example at the input we have the
input field is i will write it in one
day so we don't write too much the input
field is simply the product of the
illumination field times they complex
transmit eva t of the of the input
transparency i will do everything for
partially coherent for am sorry for
spatially coherent illumination and we
know now that if the illumination is
specially incoherent if we have all of
this already solved the coherent casual
good say because you can simply take the
modulus square of the point function and
the autocorrelation of the coherent
amplitude transfer function and then we
can compute the corresponding quantities
for the incoherent case so the coherent
is the one who always start with ok so
do that
now we can calculate for example the
field immediately to the left of the
lens so let's call it the G sub L minus
4 left and I think I still use X double
prime is my coordinate so this is now a
a flannel propagated version of the
input field so we're at out I will write
out the big bad flannel integral e to
the i 2 pi z 1 over lambda pi lambda z 1
so this is the prequel and then we write
our big integral here g sub in of x and
then the quadratic and of course two
things happen at the lens plane first of
all is the lens itself that will impose
an additional quadratic phase delay upon
the signal but now it's a little bit
different than the geometrical optics
case i forgot to mention earlier in that
i have added a thin transparency in
front of the lens so this is again a
pupil mask it does as we will see in a
moment it does pretty much the same job
as the pupil mask did that the fourier
plane of a 4f system but in this case we
actually put it a sort of in contact
with the lens in fact we don't have to
we can put the mask and what we like but
anyway it makes the math a little bit
simpler than this case and in most
implementations for practical issues
they also do it this way because it is
much better to have everything stacked
together as opposed to having different
elements so to floatin around in an
optical system ok so this means then
that the field to the right-hand side of
the lens so the plasma means to the
right still X double prime coordinate is
going to equal to the field that was on
the left x 22 functions now one is that
in transparency itself which denoted
there is Z sub PM PM of course stands
for pupil mask like in the previous
cases and then times another quadratic
complex quadratic exponential that
corresponds to the lens itself so this
would be e to the minus I pi X double
prime square upon lambda F where F is
the focal length the focal length of the
lens and finally the field at the output
now the output coordinate is X prime is
going to equal another frontal
propagation from the pupil mask to the
output plane so i have to write another
a propagation term here this would be
easy to this time and what I will put
now is GL + because this is the film to
the right of the of the composite lens
and pupil mask so that will be X double
Prime and then another quadratic
exponential
okay so this is it and and when you plug
in all this all this math and you this
is something we've done already so I
will not do it in the class again but
there is a supplement there's a scan of
my handwritten notes in the website what
you will see a little bit more detail on
how I did the derivation here but I will
skip some intermediate steps and I will
show the result here so the way you
handle this kind of as you imagine when
you combine all of these integrals you
will get a sort of integral that
involves all these functions G sub in G
sub p.m. and so on but also it will have
a bunch of quadratic exponential so what
you do is you expand the exponents of
these Exponential's and you rearrange
terms and you write it out this way so
these are the output coordinates the
don't participate in the integration so
we can knock them out of the integral
and then of course we cannot do much
about this thing because well we don't
know what these functions are yet but we
can certainly try to see what we can do
with the remainder so what is sort of
glare in here is that that you have in
the exponent you have quadratic terms
and you also have linear terms the
linear terms we know from experience
that they lead into for a transform so
they're nice we know what to do with
them the quadratic terms they usually
correspond to nasty things like the
focus but in this case you can see that
the quadratic term is x a coefficient
here which allows us to knock it out if
we select the distances z1 and z2
judiciously and of course that is the
result of the lens law this is the same
equation that we derived using
geometrical optics and we called the
imaging condition so you can see here
that when you satisfy this image in
condition then you essentially knock out
at the focus theorem from the
diffraction integral so that is
gratifying because we actually derived a
result from geometrical optics but this
time it is
perhaps more rigorous because it's
better than electromagnetics what is the
rest now well we'll deal with the rest
in a moment but you see that there's
another another quadratic here that we
cannot quite so easily get riddle and
this quadratic actually this extra
quadratic that popped up there is
interesting because it is not predicted
by geometrical optics so this is
something that well it will happen only
with special coherent illumination and
genetical optics cannot predict it then
actually cannot deal with it and in fact
they're not so long ago about the time I
was born in the early 70s this was a
topic of results what to do about this
term in fact the professor Goodman who
wrote your textbook one of his early
results as a young scientist it was too
she wrote a famous paper about this term
what was their means and he wrote it I
believe you know that in 1971 exactly a
level born I have not at all so so so
guesses you can still call it a
relatively recent recent development so
okay so so again we're talking about
this term over here what to do about
this one and then of course this is a
this entire section in Goodman that
talks about this term and I would
encourage you to read this action I will
show a summary here ok something
something got just messed up with my
animations here this was not supposed to
be here anyway so there is a so Goodman
actually he goes over three methods that
you can use to eliminate this unwanted
quadratic one of them is you lay out the
intro transparency as a sphere or the
spherical surface instead of the typical
planar surface that may sound a little
bit strange and is probably very
difficult to to implement in practice
but conceptually you can see what's
going on because now these points in the
transparency start with an original
phase delay you can work out the
carriage through here in fact the
conversion has to be exactly z1 the
radius of curvature so that it cancels
this unwanted quadratic so that's one
way the second way is to actually stick
a lens in front of the transparency this
is actually very often done in
geometrical geometric optics as well
this is called the condenser lens so for
example we don't have this anymore but
actually we don't have one in the
classroom we used to use these
projectors basically if you look at the
top surface of this projector I don't
know if I can take it out here but but
if you look at it it's actually lends
after the class come over and look there
is grooves on it and the grooves are
basically sort of a it's the grooves
implement equivalent of a land surface
if you take if you do the operation
modulo 2 pi on the on the on the surface
of the lens instead of instead of having
this surface that we typically think of
as a lens if you do a module 2 pi PI
ratio you will get something like this
of course
okay so that's a way to make it thin
lens and that is called the condenser
the result we put it in a projector is
not to eliminate this quadratic factor
it is simply because it makes the
illumination cohere uniform so when you
project the slide on the I don't know
you guys are maybe too young to have
ever seen anything like this in
operation but the next time if I
remember they'll bring an old-fashioned
transparency and I will show you how
people of my age used to give talks at
conferences when we didn't have
powerpoint but but but anyway so so so
this is done actually very commonly to
stick a lens in front of the just
attached to a thin object but then some
cases cannot be now for example in a
microscope it is very difficult to
imagine sticking a lens behind the glass
slide so so in some cases this is
applicable in some it is not anyway a
big application of this type of lens is
of course in overhead projectors does
anybody know another application where
people use this kind of lenses or why
what might motivate you to make this
kind of lens can attend these lessons
are commonly used actually you know yeah
see sometimes stick them on the backs of
like the ends or cars to like make a
wide angle that's right he doesn't have
application exactly to to improve the
the rear view of the driver and that's a
good one I was a thing of that yet
another one is in very high power
illuminators which are most commonly
used in photography and shooting movies
because they the glasses of as you know
is not a very good conductor so so so it
would heat up very badly if you you know
actually me live with very high power
and so for this the call actually babies
this really big light bulbs that they
use when they suit movies so they
actually use this rental as
called the friend L lens the same
phrenology in flannel propagation so
they use different lenses in these
so-called babies all right there in
lighthouse is also yeah so again well
every time that you have a really should
light source of course a matter of
practicality also right if you have a
major lens as you can imagine if the
diameter is weak then also this size
would also be very big right so in a
lighthouse you might have a lens that is
you know maybe one or two meters in
diameter then it would have to balance
over another half meter so that's not
very practical it is heavy fragile blah
blah blah so they make this foreigner
lenses to get around this problem okay a
good a good thing to remember about
those is that they generally they give
very poor image quality so if you look
if you try to use this as an imaging
lens it to you typically get a very bad
blur and very bad image quality but
obviously in a lighthouse you don't care
in an illuminator you don't care and
also the projector it is not used as an
imaging lens the image in Lance's area
lens is this one this is simply
collimating the illumination so you get
a uniform image at at the image plane so
in for jobs like this one where you
basically want to take a light bulb that
is highly non uniform it has filaments
all kinds of crap right so it produces a
very non-uniform field this type of lens
is very good to uniform eyes the
intensity and then of course if you need
to image a sharp object then you need
the real lens you cannot get around that
okay that's a bit about friend lenses
and so that's a bit of a detour that you
know but i wanted to point out some
practical ways that you might be able to
do this kind of thing if you need to get
rid of the lens and finally what the job
would not prove the back when he was
young is that you can also raise a
condition that you can neglect this
nasty quadratic again we're talking
about the quadratic that you see at the
top of the slide
so it is out if the field of the object
the field is typically when we say the
field sometimes in dramatical optics
women the size of this so if the field
is smaller than about a quarter of the
imaging lens then it turns out again
that the effects of this lens are
negligible and the weekend busy and so
the effects of this term are negligible
and therefore we can get rid of it in
clear conscience so I will not go into
the details there's a desert discussion
of the book and also a reference 303
from the book is an article that Goodman
wrote back then and he she goes into
this in a lil bit more detail it's a
pretty interesting article if you like I
can or maybe live in positive the
website if you like it's a pretty
indistinctly okay so assuming vendor
that will get rid of this extra
quadratic let me go back one so we got
rid of one quadratic from the lens law
and as I argued I spent a lot of time
arguing about this quadrant the
quadratic that maybe who can get rid of
that in as well assuming we can get rid
of that as well well anyway even if we
don't get rid of that what is left in
here is actually full transform I think
I pointed that out before if we
transformed why because you have linear
now linear exponents times some
coefficient and in here you have the
pupil mask so again you recognize
something familiar the same thing
happened in the case of the four of
system we again got the term like this
one but it was slightly different
instead of z1 z2 in the denominators
here it had f1 and f2 the two focal
lengths here we have the two distances
but it's pretty much the same term and
again the pupil mask appeared in the
kernel here so so this is a fitness
forum and of course we get rid of this
term and
let me jump through these animations now
okay so if you do not then what we
finally get is this expression here
where again this is the for dance form
of the pupil mask I wrote it in a form
that looks a little bit more like a full
dance form and of course you and we are
now arbitrary arguments these are the
special frequencies in the full
transform but what actually goes in
there is there is the these parameters
over here so basically we get again a
call a time that looks like a
convolution so we see that the image the
complex optical field that the image is
a convolution of the input field times
something that we call then again we
will call the point spread function and
this point first function just as in the
case of the forest system again here the
point spread function is given as the
Fourier transform of the pupil of the
pupil mask scale with appropriate
coordinates ok and there is a various
scale in forms and factors here for
example this one again ensures energy
conservation which is kind of important
but very often in optics who don't care
so much about the absolute intensities
which means that we don't really care
about this term what we really care
about is about a special distribution
like a type of a solvent whether
features make it to the image or they
get lost due to special filter and so on
so forth so very often this is a
multiplicative constant we just drop
them that sort of the convention now to
to get a little more insight out of this
imagine for a moment that they
m phosphate function becomes extremely
narrow it becomes a Delta a delta
function so if you substitute the Delta
function into the expression head before
then you get this expression which is of
course again very gratifying because now
the output really looks like again some
multiplicative term but then it really
looks like the input except then put now
the coefficient I mean the coordinates
have been scale so therefore you can
immediately see that this term is the
magnification the in fact the lateral
magnification of the system is given by
this expression so we basically
reconstructed or i should say they
derived the same results that we got
from geometrical optics namely the lens
law that is the image and condition the
magnification and so on and so forth now
we got them back from our from our wave
optics approach but we have to do some
approximations rate for example who had
to pretend that the point spread
function is a delta function in order to
get a geometrical optics result if it's
not which is never the case a delta
function is a very crude approximation
so if it's not a delta function then the
result is not exactly like this but it
is a convolution that is basically this
will become blurry like pepper showed
earlier you will get a blurred version
or in general especially filtered
version of the original input will
actually survive through to the output
okay and what I would like to point out
now this is something that I believe
sibekk also showed last time the same
slide what a why I would like to do is
basically pay a little bit attention to
the scaling factors here this scaling
factors are actually very important so I
mentioned that we drop multiplicative
factors that appear in the front here
this it is okay to drop but these
scaling factors that go inside the
argument is a very important because
they determine the amount of special
filtering that goes on inside the system
so so what I did
what i will do actually for a while Lisa
will compare the 4f system with a single
lens imaging system so the scaling
factors again they're different you see
f1 is inside the transfer function and
z1 in the case of the single lens so
what is the effect of that one also in
the incoherent case of course in the
incoherent case the transfer function is
the autocorrelation of the coherent
transfer function and since the
coherence function is basically
proportional to the pupil mask then the
what we call the optical transfer
function the incoherent case is the
autocorrelation of the pupil mask itself
this is this is again a very basic
result but again be careful when we
compute these Auto correlations we have
to apply different scaling factors in
the arguments okay so i will show in a
second how this works and i will show it
in two cases one is the what we call
before the zanuck a face mask actually
this is not exactly accurate there naked
did not invent exactly this invented
another one that has PI over 2 phase
shifts at the edges explicating but
anyway the function is the same so so
everybody refers to all to all of these
types of masks as zenica nowadays okay
so this is familiar we saw a few
examples in the past when it was in the
middle in the focal plane of a 4f system
and this is the case where we put it as
a pupil mask in a single line system the
same ask but it will it will do a very
similar thing but with a subtle
difference so this is what I want to
point out so this is the subtle
difference what you see here is of
course a schematic so you see it is
opaque outside some aperture then inside
you have this little extra phase delay
near the optical axis in a small region
and okay this is the mathematical
expression that I don't want to dwell
upon and there are the focus on which
plots here so these two plots are the
real I'm sorry are the magnitude and the
phase of the mask
so the magnitude is the top of the blue
plot it goes from 0 to 1 so it is one
within the Apple too so to see if that's
correct the opportunity has a size of
one centimeter so indeed this is one
between minus point 5 and 0.5 and the
phase well the face nobody knows what
the phase is where the magnitude is 0 we
cannot define the face for complex
number that equals 0 nevertheless Allah
set it equal to 0 but what I want
emphasize is that within the pass band
of the system the phase is 0 except for
a small chunk of width of point 2
centimeters where the face jumps to PI
over 20 k pi over two is of course hi
and this is the real and imaginary parts
these two are actually equivalent so the
real part it is one except at the little
extra fries protrusion there where the
real part goes to zero and the imaginary
part becomes one because of course when
the phase equals PI over 2 then the
actual complex amplitude equals I so
that's so that's what you see here ok so
this these two pairs describe both the
face mask so there'll be two transfer
functions of course are the same okay I
only plotted them here as a real and
imaginary part but have the same shape
what is really different and it
unfortunately my plot it will be too
small scale too small here you cannot
see very well is that they actually have
different sizes this one if you do the
scaling factors here for the numbers
that I ruled I think I used yeah so I
use the lambda equals 1 micron No yay
lambda equals 1 micron f 1 equals 10
centimeters f 2 equals 1 centimeter for
the 4f system and then Z 1 equals 11
centimeters
z 2 equals 1 point 1 centimeter and f 1
equals still 10 centimeters for the
single end and I work those out so that
in both cases you get the magnification
of a factor of 10 okay that's why the
numbers a little bit strange there so in
both cases you get to the magnification
fire of the factor of 10 and did it
deliberately because the two systems
give us the same effect kind of but you
can see in terms of geometrical optics
they identical they both give you a de
magnification by a factor of 10 but you
see here that in terms of wave optics
are slightly different because one of
them the forest system is scaled by the
focal length its spatial frequency
extent goes from minus fifty to fifty
inverse millimeters this is what you see
here whereas in this case it goes from
approximately minus 48 248 inverse
millimeters so the phone so the single
lens actually does a slightly more
severe special filter than the case of
the of them of the photo op system that
is to be expected of called because the
distance the one you can see it from
here the distance is the one that you
subtract from the object to the aperture
is longer so therefore this system is
cutting off more angles or more special
frequencies than this system we will see
we will see that in more detail in a
second the thing I want to point out
here is that then if you take the
autocorrelation of this function then
you get which is the coherent transfer
function or the atf if you take it to
the correlation you obtain the OTF which
is as sibekk demonstrated last time it
is the transfer function for spatially
incoherent illumination so again this is
a little bit of an exercise that I will
not do here but it is in in the notes it
is in I have posted another set of
practice problems so it is in pages 16
and 17 of the last set of practice
problems I have gone ahead and derived
this one and I would encourage you to go
through it's a bit of
it's a limit of algebra but you can
think of it as mental fitness right
because okay you might say why do I need
to do all this algebra when would they
ever have to to do so much algebra i can
just plug it into MATLAB and I get it
well I'm sure all of you do some kind of
fitness you go to the gym right when you
do bench presses what is the probability
that you need to do a bench press motion
in real life is actually very small
right but you do bench presses in order
to keep fit so there isn't would do
these kind of calculations is it's the
equivalent of mental bench presses right
so I would encourage you to do it
because presumably you are at MIT
because you have you want to have mental
fitness as well as physical fitness okay
let me let me talk a little bit more
about about these scaling factors so so
in this case this is just a clear
aperture that i put here and and you can
see that in the case of the clear
aperture it's actually very easy to do
the sort of the rate diagrams and you
can see also hear that why the well let
me back up for a second so you remember
from geometrical optics who have the
definition of the numerical aperture so
we define the numerical aperture as the
angle that we subtract towards the
optical system if you place a point
source on axis right so therefore the
numerical aperture is limited by one of
the physical apertures that are in the
system so the case of the forex system
that would be the pupil mask at some
point in any physical system the pupil
mask will cannot be infinite right it
has to be finite so the physical
diameter of the pupil mask assuming the
lenses that sells a large dinner the
pupil mask will become the limiting
factor and so we can compute the
numerical aperture then it is the ratio
of the radius of the mask over the focal
length F 1 you can see it very easily
from this triangle over here and of
course this is an approximate expression
in reality the numerical aperture should
be the
the sign of the inverse tangent of this
quantity but of course we're doing
paraxial approximations here so we can
drop the trigonometric functions in the
case of the single lens the numerical
aperture is again you can get it from
this triangle over here but now the one
of the orthogonal sides is z1 not f1 so
you can see that the numerical aperture
is r / z 1 and of course you know if the
system is supposed to produce a real
image as opposed to a virtual image then
z1 must be longer than f1 you can see it
very clearly from the emergent condition
1 over Z 1 plus 1 over Z 2 equals 1 over
F if you want z2 to be positive is so
for z2 to be positive it means have a
real image and for this it is required
that the one is bigger than F okay so
because of that you can see that if you
have the same size pugil mask the single
end system would have a smaller
numerical aperture that is sort of an
unfortunate fact of life and so another
thing I want to point out here is that
the numerical aperture actually changes
when you go to the second leg of the
system because of course the system has
angular magnification so what started as
a numerical aperture at the input it is
of course an angle right you if you take
the marriage in LA it is propagating at
a given angle with respect to the
optical axis by the time it comes out
this angle will have changed by how much
by the angular magnification of the
system so therefore the numerical
aperture that the output is not the same
it equals the numeric the numerical
aperture at the input times the angular
magnification of the system so this is
not the cause for confusion we can use
either one we will get the same actually
the same conclusions whether we use the
numerical aperture at the input or the
numerical aperture at the output but we
have to be a little bit careful that so
we don't confuse things as long as we
remember
this simple in relationship that the two
are connected by the angular
magnification so there isn't a numerical
aperture is so important is because if
you consider a circular of course you
cannot see the circle here this is just
the projection but imagine that i have
is fair circular pupil and then recall
that the point spread function of the
system is actually the full transform of
that circular pupil because this is a
clear aperture now and we learned some
time ago I don't remember one but we
learned that the full transform of
circular function is this crazy dink
so-called function that is given by a
ratio of a bessel function to its
argument and it is probably better to
think of it as a plot like this one that
has a main lobe and then a smaller side
lobe and actually it has a lot of logs
if it continues on forever but the size
of the logs decays away how fast it
decays away is one over the argument ok
so what I want to point out here is that
in both cases the point spread function
of the system looks like this zinc
functions sometimes it's also known as
an airy disk not an airy function by the
way Eddie function is different if this
one is referred to as airy disk if
you're curious what any function is you
can open the table of formulas
Abramowitz and stegun and you see a
monstrous thing that's called very
function and this turns out to be a
special case of an airy function but
anyway this is the Arab disk and what I
want to emphasize is that you in both
cases you get the the same shape of
added disk but different size you get
different size because in the case of
the 4f system the size of the aperture
the size I'm sorry the size of the
aperture is the same but because the
scaling factor is smaller it is f1 then
you actually get a bigger atf the size
of the disk in there in the frequency
domain is bigger therefore it will give
you a slap
narrower point spread function okay if
you work out these ratios over here and
you work out these coefficients you will
see that very clearly but but I want you
to get it sort of intuitively but using
the scale in theorem a for trans form in
both cases you start with the same
physical aperture but what matters is
not the physical size of the aperture
but the numerical aperture so in one
case you have this numerical aperture
this is a fun this is the physical size
and numerical aperture is our over f1 in
the other case you have again the same
aperture but now you have a longer
distance here this is still our so now
the numerical aperture would be r / z 1
so the same physical aperture in the two
cases will actually give you a different
size in the atf this will have a bigger
atf so the size of the atf will be if
you work it out it will be proportional
to actually 1 over lambda f 1 and the
size of the atf here i am exaggerating
of course will be proportional to 1 over
lambda z 1 okay so since we got a
smaller size of the atf in this case it
means that we'll get a broader psf so
which system is better well obviously
this one right because this one will
give you a narrower psf therefore it
will give you a smaller blur so of
course there's a caveat I sort of took
it for granted that the desirable in an
optical system is to minimize the blur
which in most cases it's true if for
some reason someone asked you
deliberately to produce a system that
causes a lot of blur then of course you
would go for this one but in most cases
we try to minimize blur so this means
given our resources that is given our
physical size of the numerical
we should try to maximize the numerical
aperture and that is what the father
system is doing so there's a few
comments about the resolution that they
ran the rest of the slides I don't want
to destroy the rest of your morning but
because the psf is finite you can
imagine that you if you have to point
sources that set it distance so now this
is like an experiment i'm moving to
point sources together there will come a
point where the two point sources will
match and if this is your image now you
don't really know whether you had one
point source or two of course you say
well you also get twice the intensity
that's true but very often you don't
know the intensity you start to do it
for example if you're looking at the sky
and you're looking at the bright dot and
your turn to decide is it one star or
two stars well so far we cannot yet go
to the stars and measure their
brightness right we can only measure the
total brightness that we receive here so
therefore if you're looking at the
telescope and you see this you don't
know if they started this one star of
two stars that are too close to be
resolved by yer by your telescope so
actually this sort of situation was
dealt with by a fellow called Rayleigh
and of course the size of the psf as a
worked out before it depends on the
numerical aperture so so I will cover
this in more detail later but this is
just a preview that the numerical
aperture actually gives you an idea
actually believe also professor separate
mentioned it in one of the path lectures
the numerical aperture gives you an idea
about the capability of your system to
resolve point images so imagine that
each one of these logs here corresponds
to the point spread function of one
point object so this case that well
resolved because i spaced them so that
the diameter of the main lobe falls
exactly on one null of the other log so
if you work it out and you take into
care into the where the zeros
of the Bessel function are located and
so on you cannot with an expression that
looks like this the space in between the
two sources is one point 22 times the
wavelength / the numerical aperture and
of course this would be at the input
plane if you are so this would be the
space in that the input required so that
the two sources can be resolved but when
you look at the output the distance of
course would be given by a similar
expression but the numerical aperture at
the output would appear in this case and
of course this is sort of the most
common definition some books define the
resolution instead of the diameter they
use the radius so they come up with an
expression that is exactly one half
which one is correct it actually depends
on your requirements if you require a
very crisp image or if you're doing
dealing with a very noisy situation then
you go for this one because as you can
imagine if you had noise into this one
then it becomes progressively more
difficult to resolve the two spots so in
a very noisy environment when you have a
very little light then you go for this
definition if you have plenty of light
then you can possibly resolve to sources
that are very close like this right here
you see we have very poor contrast you
basically have to rely in order to
resolve the two sources you have to rely
on the slippery dip intensity so if your
signal is noisy this deep may be lost
and then you cannot resolve any more
okay so that's a preview of resolution
we'll talk about this a little bit more