Optics (Lec - 6) - Electronics & Communication Engineering Video Lecture - Electronics and Communication Engineering (ECE)

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there's some comments from the grader
who look at your homeworks so a couple
of a couple of points to remember that
they seem to be sort of pervasive in
several homeworks one is that one will
do Snell's law we have to apply the rule
their own goals like n sine theta equals
and prime sine theta Prime right so
remember that theta prime is measured
with respect to the normal to the
interface so if this line is the
interface I have for example air and
glass here oh I think my hand is
confusing maybe I should turn off the
coffee okay so if this is the interface
aring glass yeah I think my hand was
professional okay and there a is
incident for example like this then we
apply Snell's law with respect to the
normal so this is the angle theta naught
this one out here not the other way
around but this one and of course the
same goes to the other side of the
whoops the other side of the interface
that I would go in kind of like this
that would be set a prime okay so that
was one one comment from the homeworks
the other comment is about PIR the IR
actually happens for all angle of
incidence that are larger than the
critical angle there seem to be some
confusion that somehow dar happens only
at the critical angle that is not true
in fact strictly speaking at exactly the
critical angle TIR does not happen what
does happen is this surface wave that I
mentioned in the note but for all angles
that are greater than the critical angle
this is when we have the TIR
addition for all those angles from
critical all the way to whatever it is
90 90 degrees I suppose the third
comment and actually this is the last
one
the third comment is about the speed of
light some of you and actually this is
also in relation to the swimmer problem
there was some confusion where the speed
of light is proportional to ending to
the index of refraction or inversely
proportional it is in fact inversely
proportional the speed of light goes
like C / n okay where n is the index of
refraction and I don't have the same
mnemonic for this but but one way to
remember it is that C the speed of light
is value is the highest speed that
anything can have
so therefore when light enters the
material can only go slower so so then
the index of refraction tells you the
amount by which the speed of light
becomes slower in a given and given
dielectric okay so these are these are
some things that are useful to keep in
mind as we move forward
okay these are two dots don't mean
anything it's just the ink from the
market that penetrated the piece of
paper so please ignore them
okay so last time the topic was
composite optical elements so we started
with an example of two lenses two thin
lenses that were in tandem and we did
the imaging condition magnification and
all those beautiful things with it and
then we also looked at the thin lens I'm
sorry at the thick lens where the
thickness is defined as you recall it is
defined relative to the paraxial
approximation if the distance between
the two curved surfaces exceeds the
cosine theta whatever it is that the
paraxial approximation limits for the
for the negligible distance then we call
the lens thick and we saw that there is
some differences for example the focal
length is a little bit different the
formula than it is for the case of the
thin lens but the primary reason why we
did it is because I wanted to define
this concept of principle planes so the
principle planes are moved slowly so the
camera can follow me so the principle
planes are basically look the locations
where the Rays bend at once so in a
thick optical element whether it is a
thick land so it is a composite of
several lenses in tandem as you can
imagine the Rays will Bend several times
they may Bend up down straight and so on
and so forth but the idea is that if you
take parallel ray from infinity after
however many times it bends in this case
it bends twice once here and once here
eventually it comes out at the final
propagation and so the principal plane
is the plane that you obtain if you
extend the incoming ray you extend the
outgoing ray and you find where they
meet that is the principal plane in fact
in general it is a principal surface if
you really do the geometry right
it is not a plane it turns out to be a
curved surface but within the paraxial
approximation it is a plane so that's
why we refer it with sometimes as a
principal plane or the principal surface
I kind of try to stick to the term
principal plane in the note and then
relative to the principal plane as we
did last time you can define the various
quantities the effective focal length is
simply the distance from that principal
plane to the focal point so basically as
far as a laser that are coming from the
left are concerned the principal plane
is acting like a thin lens right because
the is acting like a thin lens with the
focal length equal to the effective
focal length of the composite okay
because this is the Ray coming from
infinity and when it comes out it is as
if I had put a thin lens in this
particular location the opposite is true
for images at infinity so if you take
away from this point here it actually
goes to the first principal plane and
then takes off to infinity now of course
again this is what I get when I take the
incoming ray and the outgoing ray and I
find where they meet in general this Ray
also suffers multiple refraction but if
I take the first one and the very last
one and then I can have a thin lens at
the first principal plane so this is the
this is the idea and we can see the
reason why they are convenient when we
do the imaging condition for the day for
the composite optical element so will
not do this against a back went through
it in great detail last time but you can
see that basically if you know the
principal planes then we don't have to
worry about the rest of the physical
system anymore because we can discover
things like imaging condition
magnification and so on and so forth
simply by applying the rules that we
learned in the case of the thin lens so
for example in this case if in order to
find the image I take a ray that is
horizontal
therefore it could have come from
infinity
I drag it all the way to the second
principal plane then I know it has to go
through the focus okay and then I extend
it further then I take another one that
goes for the first focus then I know
that the first principle plane this ray
will have to become horizontal and again
take off towards infinity
so therefore this ray goes like this and
then where they meet is actually the
location of the image so this is the
ideal and then all the rest of the
formulas about the imaging distances the
magnifications lateral angular and so on
and so forth
they all fall out from the similar
triangles in this diagram just in the
same way we did them for the Finland
except now we have this sort of
additional space between the two
principal planes so you have to be
careful when we deal with the Ray that
is coming horizontal at the input it has
to bend at the second principal plane
and when we take the Ray that goes
through the first focal point and comes
out horizontal that one has to go
through the first principal plane and
the reason I'm emphasizing this is
because this is the generic edge were
the first principal plane is to the left
the second is to the right but there is
optical systems where they may actually
be swapped you may have the second
principal plane to the left of the first
principal plane there is no rule that
forbids that that can happen so that's
fine we still have to apply the same
rule but we have to be careful where we
bend the Ray so one of the homeworks you
will see the homework that we posted
today and they're due
actually next week you will see that
this happens in one of the optical
systems that will give you their the
second principal plane is actually to
the left so I'll let you work that out
by yourself but but keep that in mind
always identify which principal plane
you are using for any given imaging
condition
okay any questions about all of this
and someone say at least good morning so
now if our sound has been restored hello
hello can you hear us okay there nobody
wants to say good night
no there's no sound against okay I'll
keep going and hopefully sometime at
some point we'll have sound or at some
time you'll wake up enough to be willing
to say good morning okay so today we
will deal with a topic called
stops and apertures and basically we'll
spend the rest of the hour learning the
meaning of all these terms so quite a
few of them aperture pupil entrance/exit
popular then who have filled stopped
window entrance and exit and so on so
we'll spend some time learning what
these things are and why they're useful
in an optical system so start from the
first one they happen to stop the
aperture stop has to do with the fact
that any optical instrument in practice
has a finite size and therefore okay let
me back up so the instrument has a
finite size but a point source as we
learned it actually at least an ideal
source it emits kind of isotropically
all the way for pi steady steady ins
around the location of the point source
so what this means then is that the
optical instrument can only capture a
fraction of the light that this that any
given point source is emitting so the
aperture actually helps us quantify how
much is the light that an optical
instrument can admit given the location
of a point source so it is defined
actually with respect to a point source
that is located at the optical axis that
is at the center where all the optics
are rotationally symmetric about and
there is two concepts here
one is the physical stop so the physical
stop is you can think of it as the rim
of the optics the the physical limit the
Y on which the rays that are coming from
the source they miss the optical
elements in the system those of you who
have cameras you're also familiar with
the with physical stops that in
relatively complicated optical systems
for example the lens that goes with the
camera inside there is a little hole
that is very often adjustable depending
on the light level and so on this hole
nowadays automatically in the old days
with photographers having manual control
you could actually open and close that
opening to admit more or less light this
is called the aperture stop in the we
also are familiar with something else
the pupil of our eyes if you look at
someone's eyes there is a little hole
and if you look in a bright environment
this hole is relatively small if you
move to a dark environment our eyes have
a mechanism that kind of measures the
ambient light and opens up or close down
the pupil so this is the aperture stop
for the optical system of our high so
the Apple to stop then is the physical
element that limits the acceptance of
the light rays okay so in this case the
Apple to stop you can think of this this
sort of black thick line here you can
think of it as a physical for example a
metal plate with a hole in the center so
that hole would be the actual Apple to
stop in this particular instrument the
numerical aperture is a mathematical
quantity it is the angle subtended
from the point source to the rim of the
physical aperture stop and I was a
little bit sloppy when I called the
angle it is not the actual definition is
the sine of that angle multiplied by the
index of refraction now we've already
seen since Niles law several times you
can guess immediately the reason people
define the numerical aperture this way
is solid
this quantity is concerned when you go
through a flat refractive interfaces so
that is very convenient because well as
we'll see later the numerical aperture
is a very important quantity in
characterize in an optical system so
therefore by sticking the index of
refraction in this definition here who
make sure that the numerical aperture is
preserved as Ray's go through various
elements in the system at least the
elements without power without optical
power that is as they go through flat
interfaces so that is definition now of
course in many optical systems the index
where the object lives is air so
therefore n is 1 and also in this class
at least we deal with the paraxial
approximation where the sine of the
angle is approximately equal to the
angle itself so therefore indeed the
numerical aperture is approximately
involved equal to the angle itself
however as you have seen from the
homework if you already studied in many
optical systems the object lives in
space of refractive index different than
one for example in immersion microscopy
we put a droplet of oil we drop with
rapid ablative oil on top of the sample
then we stick the objective so that
there is oil in this space here between
the sample of the objective well in that
case we really should include the index
of refraction in the definition of the
optic of the numerical aperture so of
course you cannot speak yet but I do
want to ask a question myself now and if
you can answer oh well those of you can
answer easily those of you in Boston if
you want answer just please wave and and
we'll figure out how to do it but the
question I wanted to ask what do you
think the value of the numerical
aperture is is it smaller than one
bigger than one what is it do this if
you think that the numerical aperture is
bigger than one do this if you think
that the numerical aperture is always
less than one
what's the verdict always less than 1
okay
anybody who thinks the opposite that the
numerical aperture can be bigger than 1
this is really strange I feel like
working as ported to a different century
now ok so actually in principle and in
fact people do it I can construct an
optical instrument with a relatively
large acceptance angle for example theta
may be as high as 75 degrees 80 degrees
it is not uncommon in high-end
microscope objective objective lenses to
have a very high acceptance angle of
course this would violate the paraxial
approximation but that is no problem you
know the fact that the instrument is non
production actually does not make it an
invalid instrument so it's fine all it
means is that our simple theory here
does not apply and we have to be more
sophisticated with the analysis but we
can have a large acceptance angle for
example so in that case the sine of this
angle may be close to one maybe 0.9 or
point 95 and also in addition remember
that we can use immersion so we can put
the object in liquid of index of
refraction for example if it is water
that would be 1.3 if it is one of the
emulsion oils that we'll use in
microscopy what is the index typically
1.4 1.0 just the same as glass okay so
even higher than 4 for immersion
instruments 1.5 for 5 1 4 I'm sorry 1
point 5 1 4 and so in that case if I do
one point five one four times point 9 of
course it will be bigger than 1
I don't know what it is but it would be
definitely bigger than one so actually I
can have a numerical aperture that is
bigger than one that is a kind of a
useful point to remember related to the
notes or the numerical aperture is used
very commonly in microscopy when we
characterize objective lenses typically
we characterize them by their numerical
aperture in photography especially
people prefer to use an alternate
quantity known as the F number so the F
number is kind of the inverse of the
numerical aperture times a factor of 1/2
which always confuses the hell out of me
but but anyway the F number it is
written like this and people replace the
hash symbol with the actual number for
example if you look at at the lens of a
camera it might same thing it may
something like f / 8 this means that the
F number of this camera lens is 8 so the
rule of thumb is what does your money go
when you pay money for a very good
quality microscope objective one of
others that cost $10,000 or so typically
you pay in a microscope objective you
pay for a large numerical aperture when
you walk into a store and you buy a
high-end expensive camera lens one of
those Nikon or or a Zeiss really
expensive multi-element lenses that look
about as big as you know as my computer
here what you pay there is for a really
low F number ok so generally large
numerical aperture is good or
equivalently low F number is good
provided you can afford it of course
okay so the moment the way we defined it
so far the numerical aperture it seems
to be related only to the energy
acceptance of the optical system that is
how much how many watts of optical power
the optical system admit so these are of
course important but you might wonder
why am I making such a big deal in white
pull pay so much money for high na well
again the energy acceptance as you can
imagine is incredibly important because
it will determine how many useful
photons from the object you admit into
your system for imaging purposes but as
we will learn later the numerical
aperture also defines the resolution of
an optical system and right now you
probably know from everyday life what
resolution means people talk about
resolution in their camera and your
television and so on and so forth we
will do a precise definition of
resolution later about a month from now
will define it very precisely and we'll
see why and how it is related to this
numerical aperture quantity but again
the rule is that the lie that the
largest the numerical aperture or
equivalently the smaller the f-number
the better the resolution of an optical
system so we will see this can come up
again and again okay so that's the basic
definition that we that we use now
related to this concept of a per to stop
a numerical aperture then is to sort of
derivative definitions and these are
called the pupils the entrance and exit
pupils so in order to define those I
kind of need a multi element optical
system so a sort of concocted one over
there I just made it up this is not
nothing in particular I just stuck some
lenses in tandem and I put an apple to
stop somewhere in the middle okay so
think of this again as the physical
element a hole that I put in there so
the reason we need this additional
definitions of the pupils is that if the
aperture stop is varied inside an
optical instrument as it happens for
example in your camera lens if you look
at your photographic camera that you
have at home you will see that the
aperture is buried inside a relatively
large lens barrel if you have a
sophisticated camera the aperture is a
physical element
the whole length so if you look at the
at your lens from there from the one
side that looks towards the object or
from the opposite side that looks
towards the words the CCD you will see
an opportunity but you will not see the
aperture itself you will see the image
of that aperture stop through the
elements that precede or succeeded so
because of that people have defined this
concept of the entrance and exit pupil
so the entrance pupil is what you get if
you take this aperture stop and now you
treat it as an object in the imaging
system and you imagine through the
proceeding optical elements now here
have committed a mortal violation I
emerged from right to left we said last
time that every time we will write down
an optical system the light is assumed
to go from all the ways from left to
right so here this is an exception when
we deal with a pursue stops we actually
image from right to left right because
we really need to look at this as the
object and image it through the optical
elements that precede it so if we do
that then we get the entrance pupil so
in this case it would look something
like this
so this is now not a physical physical
component anymore it is the image of the
physical component that was inside the
system and we imaged it through the
elements that preceded it in that
optical system okay similarly we can
define the exit pupil if we image the
physical apples to stop through the
elements that succeeded okay so the two
definitions are kind of conjugate okay
now with respect to this to this
entrance exit pupil egg entrance exit
pupils and aperture stops this is a
mouthful who can define two significant
rays that come up in the design of
optical systems all the time and we'll
see why
the first one is called the chief ray
and the chief ray is defined as the Ray
that goes through the center of the
upper to stop the physical aperture stop
okay so that's one the other is called
the marginal ray and it is the Ray that
goes through the rim of the physical
aperture stop that is the Ray that just
clears the aperture if you take the next
ray that would propagate at a slightly
higher angle with respect to this one
this leg would actually hit the physical
block that is located at the aperture
plane and would miss so this last ray
just before the upper two stop blocks
the rest of the light this is called the
marginal ray so you can already see the
importance of this because the marginal
a kind of defines again the angle of
acceptance of the optical system except
this time notice that I've done this for
an off-axis object point
recall that the numerical aperture the
way I define it before it was still the
angle of acceptance but for an own axis
object point here and the marginal in
the Chi phrase I can define them for any
point I like okay any point I like in
the other it is in the object space okay
and this statement is sort of okay I
already decided that they defined that
vis Ray's helped us define the angle of
acceptance
now before I show the next slide and
since they're these today is this too
important race what do you think they
might go with respect to the entrance
and exit pupils by definition they go
through the center and edge of the apple
to stop respectively the chief and
marginal ray so therefore with respect
to the entrance and exit pupils what do
you think the rays would pass
so the answer is that they should pass
through the center and edge respectively
as well because the entrance and exit
pupils are images of the aperture stop
so for example if the chief area goes to
the center of the aperture stop then you
should also go through the center of the
entrance and exit pupils because these
are images of the aperture stop through
the preceding and succeeding element and
similarly the marginal ray hits the rim
of the Apple to stop therefore it should
also hit the rim of a of the act of the
entrance and exit pupils respectively
now for the marginal rate there could be
magnifications involved so for example
the distance from the optical axis to
the rim
it may be different in the upper to stop
plane then it is for example in the
entrance pupil plane and that is because
in this case the optical system has
magnified the upper to stop so the
entrance pupil in this case is bigger
than the physical stop itself
so therefore the marginal a it hits the
optical axis at a different location and
also as you can see there's an inversion
it hits at the bottom at the Apple to
stop and then it hits at the top at the
entrance pupil this is fine but what I
because of the way optical systems
behave they can produce inversions and
magnification and as we have seen but
what I want to emphasize is that they go
through the limb because of that
property that that the aperture is
imaged at the entrance and exit pupils
respectively
okay and of course the two eyes will
also meet again at the corresponding
image point at the image plane right
because the today is emanated from the
object if the imaging system is good
that is if it does form an image
properly and we don't have some kind of
a weird
defocus or some other kind of
inconvenience then the Torah is should
better meet again at the image point
right because the entire spherical ray
bundle that emanated here that diverged
here it
also converge by the time it arrives
here so therefore these two rays are
just members of this spherical lab
bundle so they will meet again at the
object point okay so any questions I
know this frustrating even if you have a
question you cannot speak your mind but
any questions can you just wave at least
so I know to save time later who answer
the question
so how do you define the distance
between the scan here we move on with a
slightly different concept and then
we'll come back to this topic again so
the next topic is a different kind of
stop called the field stop to remember
the definition of chief ray we said that
the chief ray is a ray that goes through
the center of the pupil I mean the
entrance pupil the aperture controls
okay so here is a chief ray from a
tunnel object and here is a chief ray
from a short object the bother the chief
rays they will go through the center of
the Apple to stop by definition and of
course they will also go through the
center of the entrance pupil the
entrance pupil used to be here if you
remember I'm not drawing it anymore
because I don't want to clutter this
diagram but it used to be here so the
Torah is they meet at this point because
there are the images of the aperture
however these layers will go off access
at other points of the optical system
and there will be some point in the
optical system where one of these guys
may get cut this element is the optical
system that limits the chief rays okay
that element is called the field stop
and the physical significance of the
field stop is actually very important
and even more intuitive than the Apple
to stop the field stop is what defines
the field of view another field of view
has a formal definition it is this angle
here that they do but even in colloquial
language when we say the field of view
we know exactly what we mean in a moment
the field of view is basically the size
that can be visible at and outside the
field of view the image is not form so
an example is if I look at you well
actually this is great because I look at
you through a camera right and the way
the camera is pointed right now I can
only see the central row of desks out in
Cambridge but I cannot see those of you
who are seated on the left
the right so you might be picking your
nose for example and I would not be able
to see it and that's because you are
outside my field of view thank you for
the control for telling the camera now
what is happening my field of view is
shifting okay so the field stop in this
case is actually the dye the electronic
dye that captures the image
electronically on your side then
transmits it to us here in Singapore
this is the physical field store that
limits how many of you I can see it can
at any given instance so the field stop
is is second to the numerical aperture
and the aperture stop is another very
important quality it tells us how big is
the size of an object that we can see
okay so if you look for example at a
telescope if you look at the night sky
through a telescope the field of view is
there basically how many stars can fit
what portion of the sky you can fit
within your image when you look through
the telescope when you look to a
microscope similarly the size of the
specimen that you can observe in the
microscope that would be the field of
view now intuitively we would like to
define the field of view in physical
size in meters or millimeters or microns
or whatever the case might be for issues
that will become apparent a little bit
later we prefer to define the field of
view as an angle actually the angle
subtended towards the object and you can
guess I don't even have to make the word
you can guess why we define angry
instead of physical side that is because
very often optical optical systems has
have variable magnification so with the
same optical system might move a lens
and then the object would move but the
field of view might not change okay so
if I define it as a size and I move it
here then the size would be bigger right
whereas the angle generally is preserved
so that's why we define the field of
view as an angle in radians or degrees
instead of instead of physical distances
the meaning of the field stop
and similarly to the entrance and exit
pupils that we saw for the Apple to stop
we can define equivalent quantities for
the field stop but we'll call them
windows so if we image the field stop
through the proceeding elements we get
what is called the entrance window if we
image it through the succeed in elements
we get what is called the exit window
and generally in an optical system these
windows they should coincide with your
object and image planes respectively so
the entrance windows should coincide
with the object plane the exit window
should coincide with the image plane if
that's not the case there's something
wrong with the optical system it can
happen it is not impossible but there's
something wrong with the optical system
ok so now that we did that let's put all
these definitions together this is
another confusing diagram but it solves
everything it solves that safe and
marginal rays it solve the two physical
stops the upper two stopped in the field
stop and then it solves their images
with respect to the preceding and
succeeding elements that are called the
entrance and exit pupil and windows
respectively I realize this is a bunch
of terms that I threw at you at once
so so treat this as an information
session then go back and make sure you
read them again from the north and from
the book so you make sure they sort of
solidify in your mind but this diagram
actually spent a lot of time making this
diagram because I actually drew it by
hand instead of using an optical
software I do it by hand so took me a
lot of time to make it accurate but to
the best that I can
I tried it split the hardware except for
this lens here this less doesn't look
very good it's a negative lens it should
have sort of developed this way bangle
even more if I had done this properly
this would then become monster so I
didn't do it but anyway but you can see
how it works
here's the chief reg let's stretch it
through the system
so here's the chief ray it goes to the
center of the entrance pupil then it
goes through the edge of the field stop
so that means that I chose that see fray
the propagates at an angle equal to what
equal to the field of view right because
this is the last chief Frey admitted
through a system if I put them chief ray
at a bigger angle that is a taller
object it would miss the field stop okay
then the same chief lyric goes on it
goes through a center of the Apple to
stop again then again through the center
of the exit pupil and finally hit the
edge of the exit window which is also
the tip of the image okay that's the
chief Rendon let's go through the
marginal Ray the marginal ray will hit
the rim of the entrance pupil then we'll
hit the limb of the field stop again why
I'll give you 30 seconds to think about
it why does the marginal they hit the
rim of the field stop again
my computer has actually timer so I can
count seconds very precise oh there is
not this so you can even speak your mind
no thanks by the way for taking thanks
control anybody why does the marginal
they also go through the rim the edge of
the field stop
because the entrance window is an image
of the field stop or I guess the other
way around so the marginal rate is I me
age the interest window setting beyond
the edge of the field stop also these
two planes are images of each other so
since the marginal rain meets the chief
ray here then it should also meet that
safe ray here again another way to put
it is that this is actually an
intermediate image plane for this
particular optical system okay so then
in other words you would see an image of
the object itself here if you were to
stick your high okay then the margin leg
will go again now goes again through the
edge of the upper to stop again and that
is of course because it went through the
edge of the pupil and since the pupil of
the upper to stop our images of each
other then it has to go through the edge
again okay now we'll become repetitive
it will hit again varying of the exit
pupil for the same reason and then
finally it would hit the edge of the
exit window because the exit window is
an image of the field stop
so these multiple imaging requirements
they sort of make this all makes sense
ray but but this is how the world the
marginal ray as the name suggests hits
the rims of everything the Apple to stop
the pupils the windows and all that the
chief ray it goes through the center of
the pupils and runs an exit and the
Apple to stop and hits the edges the
rims of the field stop and the
Associated entrance and exit window and
finally another to have everything
together here is again the Tiffany the
definition of the field of view it is
the ante between the last two
she fries admitted by the system last -
meaning that it is the last - that the
field stop with admit and the numerical
aperture is the angle between the two
marginal rays
that depart from the center of the from
the center of the window that is from
the optical axis itself okay we do not
define the numerical aperture from from
off axis points the reason is that in
poorly designed optical systems let me
say in an ideal optical system the
numerical aperture should be equal no
matter where you are within the field of
view the numerical aperture should be
the same in practice that is very
difficult to do and if you do a
particularly pull a particularly poor
job as a designer the numerical aperture
can decrease drastically when you go of
access here in fact in the next example
that I will show this will happen the
numerical aperture become drastically
smaller as you go off axis that is very
that is called vignetting it is one of
those French words that have crept in
optics and is difficult to pronounce you
will see it written in a second I will
just add it down here vignette in it is
very annoying because if you have
vignette in an optical system then the
edges of the image they appear darker
remember the numerical aperture is the
amount of light that the optical system
admits well if I admit more light in the
center I have a larger numerical
aperture in the center and the America
numerical aperture decreases as a goal
of axis it means that the edges of the
image will be darker right it will be
dimmer and that is of course very
undesirable and generally when we design
microscopes telescopes and such we try
to make sure that this will not happen
but anyway because this could happen
that's why the numerical aperture is
defined at the center or on axis so that
it has an unambiguous definition okay so
having said that let's look now at an
example that will actually reveal this
vignette and phenomenon in addition to
other things so this example is
very simple it is a just a single thin
lens and then in front of the thin lens
were recruited with put an aperture stop
and you can think of it actually this is
a very simplified model of the eye of
the human eye who have a lens in front
of it in the human eye we'll call it the
pupil leg of course in the human eye in
actuality also has an additional lens in
front of the pupil called the cornea but
anyway so that's why this is perhaps not
an ideal model but but anyway that's the
example that chose to do and in addition
it has a field stop which is located at
the image plane now generally if you see
a stop at the image plane this is a good
candidate to be the field top right
because clearly if I image anything
outside this opening this is what
aperture means by the way literally
aperture means opening so if I need
image anything outside this opening then
it will hit the block and therefore it
will not damage so therefore this is
very clearly the field so physically the
field stop is for example as I mentioned
earlier it is the size of the chip that
registers the digital image right this
is a very simple instance of a field
stop in old-fashioned cameras that used
film when the field stop would be in
that case the film itself read wherever
the film is chemically active this is
where you record the image outside it
just lost whatever is imaged outside
that area is lost okay so this case it
is pretty clear who is the aperture and
who is the field stop so you can see
very clearly that if you take an object
point at the on the axis then the
physical aperture here will block the
angle of acceptance from this particular
point and therefore this is the Apple to
stop and also because there is no
elements to the left this is also the
entrance pupil right if there is no
elements to image it through the
aperture stop will also be
simultaneously the entrance pupil
what
means that if you look at them as an
instrument you will actually see the
pupil itself before you AC anything else
and so that's why the two things are
actually coincident in this case what
about the exit pupil
well the exit pupil is what I will see
on this side if I image this element
through the succeeding elements so the
succeeding element is this lens here so
what will I see if I'm at this through
the lens well this element now forget
about the actual object of the system
one will question when we ask what are
the entrance exit pupils this is now our
object we are trying to see where this
would be imaged through the optical
system and we notice that this is
between the focal point and the lens
okay here's the focal point here's the
lens this is in between therefore this
will form a virtual image so the exit
pupil would be somewhere out here it
would be actually to the left of the
lens that is not surprising all it means
that if I go here and look at the
optical instrument this way what we will
see I will see a virtual image of the
pupil very similar to the virtual images
that you saw last time in the demo in
the class when Pat brought the magnifier
objective okay so that's that's the exit
pupil over here and what about the
marginal in-chief range well here is the
here they are here's the marginal ray
here the chief ray the chief ray by
definition has to go through the center
of the Apple to stop so that that is
indeed the marginal marginal ray hits
the rim and what about the field of view
where the field of view will be defined
by the pupil AB so it would be defined
by the field stop now notice that the
field stop is also the exit window
because there's no optical element to
its right so therefore that is the exit
window the entrance window is at the
object plane right since this is the
object plane this is the image plane
satisfy an emergent condition that is if
you image the field stop through the
proceeding optical elements you will
arrive here so that's the entrance
window and of course the field of view
is defined by the tutsi phrase that just
clear the field stop so here they are
and the angle between the two is the
field of view by the way there isn't one
ambiguity in the literature some people
define the field of view as the full
angry some people define it as the half
angry it's a matter of taste
how you want to define it but just make
sure you you tell people what you really
mean if you really bring the full thing
you know okay okay and oh yes nevermind
and in this case I actually worked out
the math I will not go through it here I
will let you do it by yourself if I put
some symbols for example if I call the
the size of the aperture stop I call it
a the size of the field stop I call it W
prime then have distance Z between the
upper to stop at the lens distance s
prime between the lens and the field
stop and then this calculation shows you
where how you find the the other
elements of the system for example when
it comes to the entrance pupil what
should you do well for the entrance
pupil we have to consider this optical
system is that aperture stop then have
distance Z then have the lens and then
if well I'm sorry the entrance pupil is
the I should have said the exit pupil so
where is the exit pupil will have to
image this up or to stop through the
system so how do we image it well let's
say that the exit pupil $0.08 is at some
distance Z prime to the right of the
lens well how do we find the prime we
apply the imaging condition 1 over Z
plus 1 over Z prime equals 1 over F we
can solve that 1 over Z prime equals 1
over F minus 1 over Z this is also known
as Z minus F over F times Z or Z prime
equals F times Z over Z minus F now I
said that the the Apple to stop is wheel
in between the focal point and the lens
that means that Z is less than F in this
case therefore this quantity since Z is
less than F this quantity turn out to be
negative if it is negative it means that
my assumption before that the exit pupil
is to the right was incorrect my
assumption would have been correct if Z
prime had turn out to be a positive
quantity since this is turning out to be
negative here it means that the actual
exit pupil is to the right I'm sorry it
is to the left
opposite from my assumption so therefore
the exit pupil is over here as I already
said as I already said before and better
this result these are these are all the
quantities here you can see the marginal
way from the field edge the marginal way
from one axis the chief ray and so on
and so forth and the last comment here I
did not really prove it mathematically
you can do it if you like it's a little
bit of a more involved calculation but
even visually here you can see that the
angle between marginal rays on axis is
even in my handwritten diagram here it
is visibly
larger than the angle between the two
marginal rays of axis this means that
this particular system here is subject
to vignetting that I mentioned earlier
does the term here vignette in which
means it is not a very good way very
well designed optical system in
sophisticated optical system today's
rules that we follow in order to avoid
vignetting when I described the
microscope next week
I mean no nothing should it mean next
Wednesday in today's from today I will
show you how my microscopes are designed
in order to avoid vignette in' and
that's it i've already gone two minutes
over time I think we conclude unless
there's any question that someone asked
not whether some