Optics (Lec - 8) - Electronics & Communication Engineering Video Lecture - Electronics and Communication Engineering (ECE)

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okay so while we're waiting for the
phone I mean for the microphone and a
little bit of catching up from the
previous lecture you know on the slide
where there is a telescope so I don't
want to spend too much time on it if
you've already seen the telescope at
least in two in one homework problem of
the previous set so you're familiar with
it then we also alluded to it earlier so
it is kind of opposite than the
microscope in the sense that okay both
instruments are similar that is because
they magnify but they magnify what in
the case of microscope we see very small
detail thank you
okay we've got a new battery now so who
so we're back in business
okay so in the microscope the object
appear small because it is physically
very small and it is available nearby
that is the definition of a microscope
in the case of a telescope the object
appears small not because it is
physically small but it is but because
it is very far away so for example a
star a star is humongous and it may be
several thousand times bigger than the
earth but it appears very small because
it is also maybe thousands of
light-years away okay so this is the
situation where we typically where we
use a telescope so you see the function
of the telescope here so this is the
case of a telescope with an object at a
finite location so this case the object
is physically big it appears small
because it is far away so the telescope
in this case it is actually very similar
to the microscope the way it works it
creates an image of this object in the
following sense that each point in the
object becomes a parallel ray bundle at
the output of the telescope
so therefore the observer
recall that that the final final image
is produced by the lens of the observers
eye on retina so the eye of the observer
picks up this parallel ray bundle and
focuses it on the retina now if you
extend the Rays backwards through their
final eyepiece of the telescope you can
see that the telescope actually is
creating a virtual image that is
magnified with respect to the original
object and that's why that's how the
telescope magnifies this remote object
it is more common to use the telescope
as shown in the second case where the
object is actually at infinity and that
is the classical case of a telescope
where now instead of having spherical
waves arriving from points in the object
now I have a plane waves so to a good
approximation to an excellent
approximation stars are located at
infinity but not necessarily even
terrestrial objects that are located say
more than a hundred meters or a
kilometer away they satisfy this
approximation quite well so they produce
this almost parallel ray bundles so the
telescope in this case it is a focal in
the sense that they are to the scalar
telescope again you produce a parallel
ray bundle in the reason of course is
that you want the observer to form an
image with an accommodated eye okay that
means that still the image must be at
infinity so that the eye without any
strain the I can focus it on the retina
and see that is the case it means that
the total now let's be a little bit
careful with the terminology the total
power of the telescope is 0 that is the
telescope has no optical power its focal
length is infinity however it does have
magnifying power what a telescope does
in this case the angle between a chief
ray and the axis the telescope magnifies
this angry and this is now by definition
a magnification of the telescope and you
can see that by virtue of magnifying
this angle the telescope also magnifies
the final real image that will be
produced on the retina and in order to
be a focal this was a topic of the
homework that I gave you guys a few I
don't I think it was two weeks ago in
order to be a focal the distance between
the 2 lenses must equal the sum of the
focal lengths it is very easy to
convince yourselves in this case that
that an instrument such as this one the
in the in the image in in the matrix of
the instrument the matrix will look like
this and it will have a zero in the one
two element in the first row second
column it will have a zero if that is
the case whoops
if that is the case then it is called a
focal so by by satisfying this condition
D equals F objective plus F eyepiece we
ensure that this is the case okay so
there's a number of different types of
telescopes I just picked one here the
the textbook lists all of them there is
the cap Larian Galilean astronomical
telescope they all have different
combinations of lenses I picked one here
that is composed exclusively of mirrors
and it is very common actually in
observation in observatories it is
called the Casa Grande it is also the
same type that if you go to the
discovery store for example and buy a
telescope for your nephew or your niece
this is the type you look at it you know
that you busily entered horizontally and
you look at it from the top and this is
the schematic here the object or the the
sky is on this end and you place your
eye this way so the telescope itself
consists of a large first mirror that
functions as as the sack as the primary
the enters the second mirror so if you
look at the
this side if you look at into the
telescope you will see a small block in
the front face of the of the of the
instrument so that's the secondary
mirror so the combination of primary and
secondary of course you use two of them
to get even higher magnification and the
combination is the objective and then
there is an eyepiece which is where you
stick your eye and you see the u.s.
attainment of the sky so this is a
compound purely of mirrors so last time
we are going to call this kind of
instrument cap optic then there's
another type of telescope actually it's
not a telescope it's a kind of an method
for correcting optical systems called
Schmidt Schmidt telescope or Schmitt
optical system let me say why you use
this so you recall that the ideal mirror
surface to give perfect focusing of an
object at infinity is actually parabolic
now suppose that we cannot make a
parabola so we have to make a sphere
instead of a parabola well in that case
a little bit of thought will convince
yourself that the sphere because it is
more highly curved than the parabola
near the edges away from the paraxial
region it will actually bend the Rays
that are far from the axis the it will
they will Bend more than they should so
the result is the failure to reach
perfect focus this is what we have
already seen and we call it spherical
aberration
so Smith basically came up with an idea
to correct the spherical aberration by
sticking a slightly as a glass surface
at the front pupil here he put a glass
surface which is slightly shaped it's it
has an toroidal shape and the purpose I
don't know if you can see it in Raymond
it was it perhaps I should have
exaggerated more it is too subtle but
what it does is it takes this ray
and sends it slightly higher so the
kind of functions like a negative lens
actually you can think of it this way
with a very small power and then because
this hits the surface at a different
point with a different angle then
basically by engineering the set of the
surface here you can make sure that most
that all of the Rays will actually meet
ideally at at the focal point so this is
a very simple way to correct for this I
don't know if it is simple but anyway it
is a way to correct for spherical
aberration so this is called cut that
dioptre because it contains reflector as
well as a refractive element so
according to our terminology that's what
we call it okay so that's all I have to
say about telescopes are there any
questions or either about telescopes or
microscopes or any of the optical
systems that we saw
okay so next I'm gonna shift gears a
little bit and I'm going to expand on
the very last topic that we saw the
topic of aberration so we saw quite
extensively we mentioned several times
this spherical aberration it turns out
it is not the only one
and we'll see a number of different
types of AB array of aberrations today
so to do a little bit of organization
and classification we start by
classifying them as two types one is
chromatic and this could have also seen
in the past in the context of a prism
primarily and it is due to the fact that
the index of refraction has a variation
with wavelength so if you apply the lens
makers formula or any of those in order
to design an element with a given focal
length at a given wavelength you have no
guarantee that the focal length will be
the same at a different wavelength one
fact is called chromatic aberration
again this comes from a Greek word I had
to do this sounds like like this movie
My Big Fat Greek Wedding but this time
Greek I guess it is natural so Rama in
Latin spelled chroma with accent here it
means color in in Greek so that's where
the term chromatic comes from the second
time the second type of operation is
called geometrical and there's a long
definition here but I will go back to it
and perhaps this long definition will
make more sense when we start with
chromatic because this is easy to deal
with
okay not easy to deal with but it is
easy to understand so as I said we have
seen this type of plot before where the
horizontal axis is the wavelength the
vertical axis is the index of refraction
and forget this dashed line for a moment
just look at the solid lines so these
are the index of refraction is function
of wavelength for different glasses
you have seen this before in the context
of the prism this is of course a
disaster it means that if you have a
lens like this one which you design for
a certain focal length say somewhere in
the middle of there over the spectrum
say near the yellow then this means that
well as you can see the index in the red
wavelength for sort of normal materials
this is actually called normal
dispersion the indexes is smaller at the
red wavelength so if you remember the
lens playmakers formula
smaller index means that the focal
length is what
okay the lens makers formula says that 1
over F equals n minus 1 the difference
in radii R a so if n goes down it means
that 1 over F goes down it means that F
goes up right so it means that the focal
length is longer and this is also what
you see in this picture over here you
see that if you have a white white light
coming from infinity you see that the
red rays focus at a longer distance than
the blue rays okay so that is the that
is the evidence of of chromatic
aberration and it turns out there is a
way of correcting it which is to stick
two lenses one of them with the that is
a positive lens and there are no
negative lens and first of all they have
to the two lenses have to have slightly
different indices otherwise it wouldn't
even make sense right then this could be
just one element so they're different
this is different
dispersion probably so one can pick for
example between these glasses right you
could pick one lens out of this one
another out of this one
is Kevon designed this kind of element
and then we were to understand it is
because one of them tries to focus the
race the other because it is negative it
will try to actually defocus them you
can imagine how one can compensate the
action of the positive lens with the
action of the negative lens and
eventually cancel or at least
approximately cancel chromatic
aberration over a relatively long range
of wavelengths I used to give this
problem as a homework I don't do it
anymore but but but anyway it is a
fairly straightforward thing to do and
if you go over the textbook there is an
entire section where he tells you in
detail how this kind of thing can be
done conceptually it is very simple all
you do is you balance the focal length
so you make basically make sure that the
composite has a focal length that is the
nominal focal length and then you
balance the chromatic values of the two
over the two elements of the composite
the geometrical aberrations are more
interesting because conceptually they
are actually much richer than chromatic
and also algebraically they're much more
difficult to handle so people have come
up with all kinds of clever tricks in
order to understand and express
geometric aberration so again let me
remind you that at the exit of an
optical instrument the ideal wavefront
should be spherical so this is the
dashed line over here if you draw
normals to this spherical wavefront of
course there will be the radii of the
corresponding sphere and they will meet
up at the center of the sphere so if I
actually produce a spherical wavefront
then I can expect a perfect geometrical
focus at the center of that spherical
wavefront so the problem of course is at
optical instruments as we saw repeatedly
in a number of cases in the past
including spherical reflectors spherical
refractors and so on and so forth they
don't produce spherical wavefronts at
exit they produce wavefront that in
general may be some complicated surface
so in this case use
is of course an exaggerated I made it
terribly aberrated
and this is the solid line over here so
the the difference between the actual
aberrated wavefront and the ideal
wavefront which is of course spherical
this difference is by definition the
aberration so try to be fairly loyal
here so I again this is a cartoon but I
took the difference between the solid
line and the dashed line and this is
what you would get the two typically you
measure it in optical path length and it
doesn't have to be very dramatic the
vertical axis here can be as small as a
few wavelength and that is often
sufficient to destroy the quality of the
image formation in an optical system in
fact we the book as the story of the
Hubble telescope which is kind of very
famous or I should say infamous story of
the disaster and heroic in optical
design the telescope was sent to space
with a serious error in the design
actually in the in the assembly that
caused huge very collaboration and then
we had to send another spacecraft to fix
it and so on anyway this actually
succeeded which is impressive but anyway
so the what the book says is that the
error was actually turned out to be I
think it was half a wavelength or so
whenever that was sufficient to to
destroy the quality of the images that
were beamed from the telescope so one
has to be very very careful in this
business ok so as you can imagine if you
try to express this kind of complicated
surface or it's different from a sphere
you will probably get a very nasty
dramatic goona metric expression so for
a number of reasons one of them has to
do with the ability to compute
analytically which is kind of an
old-fashioned problem because now we're
computer so that's one reason but the
second reason because of the desire to
get intuition about how these things
work
people normally Express aberrations in a
form of a Taylor series and of course in
word even now in in this is a surface
not not not just a curve as I saw here
the Taylor series is is high dimensional
that you actually have many terms for
each order in the Taylor series
expansion so these terms are actually
grouped into in two orders and the
variations that to end the talent that
we consider as a first step in an
optical system and the third board
deliberations that are also known as
Seidel aberration and of course the the
second order reparations disappear
because we assume rotationally symmetric
optics that kills all the second order
terms of the Taylor series by definition
and the first order aberration is
actually only one and that is the focus
so first of all the reparation is
actually paraxial it is something we've
already considered and then it is the
state the third order that we deal with
in optical design so there's all totally
you go through all the symmetries and
all that it turns out that the third
order aberrations it's five of them one
who've already encountered it is
spherical and there are therefore a
comma astigmatism curvature of field and
distortion and what you see here is the
aberration wavefront so this probably
doesn't mean very much what we'll do
next is will actually see a ray diagrams
of all these aberrations and what they
look like okay so the spherical visit
diagrams are straight out of the book
who have seen actually several times and
the way spherical aberration looks like
it basically happens when the Rays that
are away from the axis they receive more
curvature than rays that are near the
axis or more or less okay so in both of
these cases you get the situation where
the Rays from away from the axis they
focus before or after the paraxial focus
it is interesting the way if you look at
this kind of system you kind of wonder
what exactly is the focal spot and
that's not an easy question to answer so
if you look at that at the marginal rate
over here the size of the beam that you
create is of course defined by the
marginal rate so you see that the
marginal a kind of focuses here then
starts to focus less then it is a
minimum size and then starts to expand
again right because now this aberrated
ray takes over so there's a minimum
width in this optical beam with this
aberrated optical beam to produce and
that is called the least the circle of
least confusion it is kind of a
compromise to see whether the system is
approximately InFocus another thing that
I want to emphasize from this diagram is
that as you can see near the edge here
of this bundle there is a congruence of
rays that approach so if you this you
can only see a few days here but if you
imagine that you fill up the space with
Ray's then you would see that you would
get a very tight intersection of
successive rays that trace a curve here
this curve is called the caustic and
again that's another Greek word
if sticky means something that burns
okay and the reason it is called caustic
or the reason it it eludes the the word
eludes to burning is because these
congruence of res if you were to put a
piece of paper there if you wanted to
put you know an element you would create
and increase the intensity of light near
the edge of this of this bundle and
therefore it could cause presumably
elevate the temperature since you have
more light energy and that might
actually result in burning caustics are
very interesting things I will say a
little bit more perhaps next time about
caustics but they occur often when you
deviate from a perfect spherical
wavefront when you pick different
reference other than spherical this
caustics have a tendency to occur and
there's a very beautiful mathematical
theory that describes that it's very bit
advanced so will not go into it is kind
of beyond the scope of this class but I
thought since also the book has
mentioned it I thought it would be
interesting to point out the existence
of this caustic when it comes to
spherical aberration it has of the
several things one can do in order to
control the spherical aberration one is
the actual shape of the lens if you go
back to the lens maker formula this one
over here I've already written you can
see very easily that you can get the
same focal length with a number of
different combinations of radii for the
front and the back surface of the lens
right so from the paraxial point of view
it makes no difference whether you pick
for example whether you pick Plano
convex or whether you pick a by convex
lens or whether you pick a meniscus and
so on from the paraxial point of view
they're all the same because as long as
you have the same focal length however
because this theory
now we've gone beyond the paraxial
approximation this theory is non
production actually the shape factor of
the lens makes a difference and then
forget the definitions here this plot is
the most interesting one what you can
see different landscapes
this is a meniscus kind of that is a
convex meniscus then this is Plano
convex this is by convex again Plano
convex but cliff the other way around
and then a concave meniscus and this
plot was computed so that all of these
different shapes they have the same
paraxial focal length but on the
vertical axis you kind of see the amount
of spherical aberration that you receive
and you can see that it is very
different of course and and it actually
has a minimum it is not quite zero but
it is a minimum and the minimum occurs
typically near the Plano convex shape
for a for an object at infinity okay so
why is that well the reason is this is
kind of nicely illustrated by another
diagram that I stole from your textbook
if you if you place the Plano convex
lens such that it's flat surface faces
the plane wave coming from infinity then
basically refraction occurs only at this
spherical interface so we have no hope
you will get very strong spherical
aberration because as we learned earlier
the spherical surface is very far from
ideal as a focusing element so this case
is very highly overrated now it can do a
very simple thing you can flip the lens
so that the spherical wave meets the
spherical I'm sorry so that the plane
wave meets the spherical surface if you
do that when you actually get refraction
to occur in two steps and because the
first step produces an aberrated
spherical wave but the second step also
encourage some spherical aberration
because of the nonlinear
pain in Snell's law it turns out that in
this case the second surface not exactly
but closely compensates the spherical
aberration and you get a much better
focusing quality over here so there's a
very good practical advice that would
give to the sort of first-year graduate
students when when they go to the laser
lab is that when you have when you try
to to focus a plane wave you always
orient better a you use a plane or
convex lens not about convex and B that
you place it in this way so that the
spherical surface of the lens meets the
planar wavefront and of course it also
goes the other way around if you have a
point source and you want to collimate
it then again you have to make sure
basically you can reverse their eyes
over here and you have to re-enter the
lens so that the planar part of the lens
faces the point source so this is an
example of doing a very simple trick in
order to not quite eliminate but but
reduce the spherical aberration it is
out yes push the bottomless TV
impossible in that graph how did they
actually measure the spherical
aberration in different landscapes yeah
so the way to thank you so they've
mentioned that so you can characterize
spherical aberration based one of these
on one of these parameters here one is
so it's called the longitudinal
spherical aberration and it is the
distance on axis between the paraxial
focus and the focal point of the last of
the marginal length right another way to
do it is to basically measure the size
of the least confusion so any of these
if you think about it all of these are
proportional to each other so you can
put any of these quantities on the
vertical axis here I'm not sure what you
did here at
it says axel here so it was it was the
longitudinal distance here and I think
you should've mentioned that this way of
characterization is common for all of
their variations that we'll see in the
next two slides yeah but when we
purchase a lens do this specific word
axial focal length or circle of fleas
envisioned as a F yeah that is a really
good question if it is a sort of
high-end lens it should be well
connected
so then what they will specify is
actually the least confusion focal
length but it is pretty close to
paraxial if it is a poor lens than most
likely what they specify is the paraxial
focus and then it is kind of up to you
to to position it and other questions
okay in the papers yes a job in the
previous sites what is a curving the
third and previous then yeah what is the
curving the third graph the curve in the
paragraph yeah so the vertical axis is
age you know horizontal axis yeah I
don't remember actually I think it is
where I was it is the amount of
spherical aberration as a function of
position in the field and not really
sure maybe you can check the actually
have the booklet you can check the book
this curve is directly from the books I
think it is the amount of a spherical
aberration it makes sense that because
any other questions
okay so it turns out that what we said
before and actually Colin made that
point when I was discussing ellipsoidal
refractors and so on and so forth what
was it before that a spherical surface
produces spherical aberration it is not
always true there is one condition
where a spherical surface does not
produce spherical aberration and this
condition is given here that's another
diagram god that I got from the book and
you can see here basically the condition
says that if the edges of the spheres
are the index of refraction of the
sphere is end to the index of refraction
outside the sphere is n1 and you place a
point source at the distance R and 1
over N 2 then of course the point source
is inside the sphere this if you think
about it this is a negative lens it will
produce a virtual image this virtual
image is located over here and it is
free of spherical aberration okay I will
not prove this this is actually in the
book it is given as an as a problem in
the textbook and the problems in the
textbook are solved at the end of the
textbook that's why I never assign
problems from the textbook and anyway
you can look at applet there is a very
simple geometrical proof of this one so
we'll skip it here but the point of this
is that is that there is actually a
single combination of image of I'm sorry
of object image pairs for which the
spherical aberration is cancelled for
this spherical surface and again a one
of a size that if you place the object
point elsewhere okay it will still form
an image in the paraxial sense this lens
will remain a lens right but it will not
be free of spherical aberration
do not it will only be free if if this
condition here is is satisfied in
actually this distance that I mentioned
is from the center of the sphere okay
all right so the reason for that so so
this now has a number of implications
for example this is one reason why
meniscus lenses are very popular in
photography or at least in valuable
fashion cameras that were very expensive
now we're talking about cameras in the
twenties and thirties you know the type
of camera that the photograph that the
cameraman would go under a black hood
going click and the family would be
sitting you know with whatever with the
children all that stuff so these cameras
they would typically use a single
meniscus lens and the reason was exactly
that the meniscus at least if you will
design it for the typical distance
remember again this was a camera for a
photography shop right so it has you can
kind of predict the location of the
object because the cameras all were
already always sitting in the same
distance from the where you place the
family to take the family picture then
in that case you can engineer the
meniscus to minimize the spherical
aberration for this particular distance
it turns out the meniscus also has some
other interesting properties but I will
not go into that okay so of course in
that case you cannot really put the
family in glass so you're going to have
exactly this situation over here but
nevertheless if you think about it you
don't need you know you can satisfy this
condition even if you have two spherical
surfaces provided that as you go from
one surface to the next you to satisfy
the same condition right it is the same
condition that you have to satisfy
applies and I forgot to mention this
condition is called the Aplin attic
condition actually that I don't know
what is called Appling attic but it
sounds like you don't call it why we
were there comes from Appling attic
means non-planar so it makes no sense at
all but anyway that's what they call it
and and it is also used in only
management microscopy in this case we
mentioned that actually it was also a
topic of one of the homeworks well
imagine means that you put liquid the
droplet between the objective and the
sample in the microscope and in this
case again you can kind of make sure
that when you apply the planet akan dish
because again in the microscope you have
very good control of the distance
between the exit pupil and the object
you you can make sure that the plan
attic condition is satisfied for this
particular distance and and you can use
the index of refraction of the oil in
order to satisfy the Atlantic condition
so now that's the case where you exactly
cancel the spherical aberration so it is
it is very interesting it is different
than the previous case here we exactly
cancel it by using this particular
condition the next variation I wanted to
discuss is common so common is different
than spherical in the sense that
spherical occurs even if you have one
axis incidence comma actually occurs or
off axis incident rays so this is the
typical image the typical sort of
depiction of comma that you see in books
again you have a plane wave coming from
infinity but it is incident at an angle
of axis so it is likely quite possible
that you might have spherical aberration
as well in this case but in this
particular case were only concerned for
the for the aberration due to the fact
that you are of axis that is called
common so if you work out the sort of
the non paraxial imagine for this case
you will discover that at the focal
plane sort of up to here what you see is
a cross-section of the of the
system in the meridional plane this is
what you would see as a front view on
the imaging plane and it turns out you
can also see it here this is perhaps a
better a better picture it looks like a
teardrop and this is where the term
comma comes from it is not a condition
that you enter after a very serious
accident but it is rather and from the
punctuation symbol comma in all fashion
books kind of the common looks like this
right so I don't know how good my
drawing is but but this this teardrop
shape is kind of reminiscent of the
punctuation comma so this is what it
looks like now the geometry of this is a
little bit complicated
you can basically calculate the shape of
the comma in the meridional plane and
the sagittal plane perpendicular to that
so the meridional plane is this one the
one that you see over here that is if
you take the line of incidence it is the
plane that cuts through the center of
the lens and then if you rotate by 90
degrees that would be the sagittal plane
so called because were off axis that you
are not symmetric anymore on axis they
are symmetric but off axis they're not
so that's what gives rise to the
strongly asymmetric nature of the comma
and it turns out if you do some simple
geometry you find out that the length of
this chromatic shape along the
tangential the original plane it turns
out to be three times the length along
the sagittal plane so this follows from
geometry to be honest I never quite went
through the derivation of this one but
anyway someone did it and this little
vessels by the way that ever way of
becoming very complicated even when they
give very simple results like this case
over here what I would like to do
there's a few things I would like to say
about common one is of course that it
also depends on the shape of the lens so
here you see
what the spherical aberration and comma
plotted for lenses of the same focal
length but but a different shape factor
and you can see that for to to sleep is
a region over here where both the
spherical actually the comma is
completely eliminated and the spherical
is close to zero so this means that this
is a good design point if you want to
simultaneously minimize common and
spherical aberration now you might
wonder let me skip for a moment the
discussion of the bottom figure over
here I'll come back to it but you may
wonder how it can be that comma is
completely completely eliminated okay it
turns out that and again this is
something I will I will give without
proof it turns out that there is a
condition that indeed you can cancel
comma so this is called the sign
condition sign now this is not a Greek
word it is actually a latin word and it
is the same sign as the sign we have in
the gamma tree and this requires a
little bit of explanation so we'll do
that and then we'll quit for today so
the same condition follows from an other
result called the optical sign theorem
in testing Li for those of you who are
mechanical engineers and nor
thermodynamics this field was proven by
a kind of a hero of thermodynamics
classes there was I think that some kind
of something called clouds use
inequality or something like that it's a
very big important result in
thermodynamics so he proved it first and
then two other people associated with
optics more closely at the and Helmholtz
they derived it independently ten years
later so what is with this theorem say
well here's again the geometry of an
optical system we use the notation alpha
as before to denote angles and why for
some reason the book uses why not
to denote the elevation of race so this
is not an exact result it is not per
axle but it says that if you satisfy the
imaging condition then the index of
refraction the Ray innovation and the
sign of the ray angle in the object
space are the same as the product of
this actually should have said the
product of the three is the same as the
product in the image space okay so you
can see this quantities over here all of
them a sub I I'm sorry this is a semi a
sum oh and so on and so forth
okay what is the significant now if you
take the ratio of Y at the image plane
so this is y sub I over Y sub all that
ratio is the magnification so you see
problem here the problem is that if you
calculate the ratio from this quantity
the ratio would turn out to be worth it
would turn out to be something like and
I
sine alpha I over an object sine alpha
object and because of the presence of
the signs here you can very easily
deduce that the magnification is not the
same
the magnification that you have depends
on the size of the object which of
course very annoying we did not see that
in the paraxial approximation but of
course this is non production that's why
it happens but it is very annoying it is
and it is the reason why karma happens
you can also see it over here you can
see a very consistent of that different
rays they attempt to reach to focus at
different locations in the image plane
so that's the reason so the sine theorem
says boy if I could actually make I'm
sorry if I could actually make this
ratio sine alpha in the object over sine
alpha and image if I could make equal to
the corresponding paraxial quantities
then it would be happy because then the
magnification would be the same in the
independent of object elevation and
therefore ax would be free of comma so
that is the sine theorem
okay so basically the reason kama can be
eliminated here it is because for this
particular lens configuration the sine
theorem is actually satisfied so you
basically get complete complete
elimination of the common okay so unless
there's any questions I think we should
quit here and we'll continue with the
remaining aberrations on Wednesday
George can I make a comment about the
I've been thinking about this
Appling assic word that you mentioned if
you go back a couple of slides to the
yeah yeah that one that though I think
that now I'm thinking where that word
comes from I've never really thought
about it before but the that of course
the appli Natick condition will be
satisfied not just at one point but on a
complete sphere with center at the
center of the spherical surface and the
perfect image will be formed on another
sphere also with the center at the
center of the spherical surface so I
think maybe that's why it says not
playing it's because the the image on a
on a spherical surface is going to be
formed perfectly on another spherical
surface we'll see