Optics (Lec - 9) - Electronics & Communication Engineering Video Lecture - Electronics and Communication Engineering (ECE)

the following content is provided under
a Creative Commons license your support
will help MIT OpenCourseWare continue to
offer high quality educational resources
for free
to make a donation or to view additional
materials from hundreds of MIT courses
visit MIT opencourseware at ocw.mit.edu
okay so I don't know if everybody's
awake but I will start so today the
topic is actually Hamiltonian optics but
we had some left over from last time
wicked so I'd like to go through that
very briefly and then I will hand over
to set back and patent who will give a
demonstration of aberrations in a sort
of experimental demonstration and also
how optical design software works and
then we'll get into Hamiltonian optics
the second hour and we see how far we go
okay so last time we talked about how
many biases we talked about spherical
common that's about it is we talked
about actually about two aberrations and
we said that there is five of them the
five so-called Seidel aberrations or
third or deliberations and today I will
cover the other three so the first one
in line is a stigmatism which is
actually for me at least it is the most
difficult to to visualize the best
schematic I have seen actually is in
your textbook it's the middle picture
over here but before we get to that and
two we see what happens is let's define
to make sure that we know that as a
terminology so astigmatism happens look
at the top left
diagram but the astigmatism happens with
object point that is off-center and this
institution of center imagine that you
know there's an fundamental asymmetry if
you take the chief ray that goes through
the through the center of the lens and
again here the assumption is that they
happened to is at the lens itself so
that chief ray goes to the
center of the lens okay so now you can
take today you can think of two marginal
rays one is the marginal rate that goes
through the original plane so the
militant a regional plane is the one
shown like this and the other is the
marginal rate that goes through the
sagittal plane perpendicular to the
original plane so segment is mockers
because the two now take cross sections
of the Rays in the two planes the
sagittal in the original plane the two
right bundles they do not necessarily
focus at the same place in the studies
and plane actually if you if you try to
visualize the center plane there's no
difference than normal incidence right
so you basically get so whereas in the
many do not claim because you are off
axis you have this property that we
discussed before that you get different
magnification at different points in the
field so as a result these two might
actually heal to different focal
distances so this is depicted here
actually where you see actually a good
political diagram you can see the
sagittal plane and the meridional plane
and you can see that the sagittal plane
actually focuses out here it is f sub s
for sagittal whereas the malignant plane
focuses here it is called F sub P for
tangential so if the if an optical
system is subject to this problem it is
called a stigmatic the typical way that
people characterize astigmatism is with
a spoke wheel and most of you must have
taken I exam the givers they give you a
similar test actually to test if you
arise at a stigmatic and you need
correction they saw your son decreased a
spoke wheel and they asked you by
basically if the spokes are
simultaneously in focus with the with
the circle they would lying the outline
circle so if so if the system is a
stigmatic then as you move a piece of
paper through the through the along the
optical axis you will see one of them
come to a focus first and the saggital
come to the focus later okay so so that
stigmatism
they hear some voices but I think that
from the next classroom not from Boston
right did anybody say anything never
okay okay
a related aberration which has to do
with the fact that well let me say first
so related aberration is called
caricature field this one is actually
easy to understand and it says the
following says that if you derive the
imaging condition for a point on axis
then you imagine you go off axis it felt
that the image may not necessarily focus
on the same plane as as the image on
axis it may focus in fact what this
aberration is about is that the image
actually comes to focus on a care of the
surface centers of the optical axis okay
now why do we care about the plane what
we want this to be a plane here well in
all fashion cameras one used to put film
at the image plane of the camera so were
to be sure that the entire image is
focused simultaneously on the film
nowadays we don't use film we use a
digital ship but in both cases there
flattery so chemic pseudo field is a
problem then because it says that
actually the off axis image points will
be out of focus because they will focus
in front of the final plane so this is a
kind of thing that if you could somehow
make care of the ship surface you could
solve it immediately all we have to do
is you know match the surface of the
chip to the surface of the the curvature
of film but in fact actually people are
trying to do something like that there's
a lot of work on flexible electronics
and that's one of the motivations but it
is not yet widely available okay so
nevertheless there is one condition
where the surface that the field comes
to focus actually becomes flood if it
was derived by a mathematician I don't
know when event was in the elderly in
1900
his name is petrol so this condition
it's actually it has a more complicated
expression but for two lenses it works
out to something like this if the the
sum of this quantities where N 1 is the
index of refraction of the lens and F 1
is the focal length so a little bit
unusual now we multiply the focal length
doing the index of refraction we're
never done that before but this is what
this guy derive if they to sum up to
zero these quantities for a pair of
lenses then the field of carpets or
vanishes and what is even more sort of
lemarchal about this remarkable about
this relation is that it does not
involve the spacing between the lenses
no matter where they are if they satisfy
this condition the field of curvature
will be will be eliminated okay and so
for example if they're composed of the
same index of refraction then the index
of refraction also drops out of this
equation and then you can treat the
magic composite as a thick lens
basically which this is a relationship
that one of the derived for for a
composite element and you can make sure
that one of them is the negative of the
other so that you can make this quantity
vanish that is F 1 is equal to minus of
2 and then you can find the focal length
from this equation over here we're kind
of cool because it says that you can
actually eliminate if it is really
important to you you can eliminate this
problem by using the Petzval condition
and again there is a more complicated
condition that applies to multi land
systems that have more than two lenses
but I didn't buy the
what is even more remarkable is that the
petrol surface and the field curvature
is also related to a stigmatism and so
what what you see in this plot here also
from the textbook is the focal surfaces
that is the surfaces in which different
points from a flat if you take a flat
object and you take different points
along this object these three surfaces
are correspondingly the tangential focus
for the stigmatic system the sagittal
focus and and finally the petrol surface
now yeah so what we discussed here is
actually the petrol condition
I haven't said you will get what is the
petrol surface okay so the person a
petrol surface is the following it turns
out that as you change the stop in an
optical system that is as you as you
move the stop in the axial direction you
can achieve a condition where the
tangential and the saggital a stigmatic
focal surfaces they collapse into one
surface this surface is called the
Petzval surface so you know not to be
confused with the Petzval condition even
though of course they are related so
what you see here is basically as you
move the stop in an optical system this
is borrowed from another textbook not
from I should have recorded the actual
visit from Jenkins and white from the
members of optics so as you move the
stop you can find the situation where
the two surfaces coincide and that is
the Petzval
surface of the system and of course this
one does not satisfy the Petzval
condition because the Petzval surface is
called is curved if I could somehow
satisfy both my pencil condition and
collapse the tooth surface is tangential
and sagittal onto the pencil then I
would have a flat a system with a flat
field which would also be free of
astigmatism
so so obviously we cannot get into too
much detail about this without becoming
quantitative and this is quite difficult
to deal with with with algebra so I'm
just being descriptive and you will see
some of this in action when subject does
the demo of the software and the lot of
distortion that they will discuss has to
do with something slightly different so
the field of curvature workers
investigate ism or curse because if you
take object points of axes they frame to
focus at the same plane they focus on
different planes and on a car on
different surfaces the sagittal and
tangential the distortion occurs because
even if you manage to focus at the one
plane the magnification is different the
magnification and this you can see very
easily actually just looking at this
diagram over here if you take the on
axis point the imaging condition and the
magnification are given by these
distances the for example the lateral
magnification is the ratio of this
distance to this distance right but what
about this point now for this point the
lateral magnification will be the ratio
of this distance to this distance and
for those of you in Boston and pointing
out to the hair to the Tilted chief ray
so I have to measure the distance from
the lens on the chief ray and you can
see very easily that this length is
different than the length of the chief
ray for the own access point so because
of that the magnification on axis is
different than the magnification of axis
therefore if I attempt to image the
checkerboard like this one is the
magnification is different I will see
this sort of of sort of I don't want to
use the word distorted you will see this
this sort of an altered let's say alter
the image and
this is the definition of distortion
Synoptics were to be a little careful
distortion colloquially means everything
let all of the stuff that they described
before in everyday English you can call
them distortions well in optics in
professional optics when we mention
distortion we automatically mean this
one you have to be careful not to use
the word distortion in the colloquial
sense right because it is reserved to
mean exactly that and I guess if it has
two forms it can be of this type where
the magnification is higher of axis that
is called pincushion or lower
that's called barrel and they believe in
this cases they yeah the top one will
give rise to barrel because in this case
the contest stop is here the Apple to
stop is here the chief ray on the object
side becomes longer of axis now if you
recall the magnification is the lateral
magnification equals s on the image side
over s on the object side okay with a
minus sign but the minus is relevant
here okay so since I made so in this
case s on the image side is
approximately the same as on axis but as
from the object side became longer by
putting the stop here so therefore I get
a lower magnification of axis there
forget this type of distortion if I put
the stop on the other side then it is
the opposite case now I make this length
longer therefore I get pincushion
distortion okay and of course a way to
eliminate distortion altogether is to
make the system like this this is
actually a telecentric system and to
make sure that the stop is exactly at
the center of the system and also yes
and
so if you do that then basically a
little bit of thought will convince you
that in this case the variation between
these lengths is actually very small so
therefore there is you know some
distortion but it is higher order the
third-order distortion is completely
eliminated here so you basically get a
distortion free image okay so before we
before I I turn over to sit back and
peplum last time I mentioned caustics I
would like you to saw what caustics are
this is actually from the last page of
today's notes which haven't done yet but
and the details here not very important
by the end of the lecture today you will
know what has happened here this is a
special type of lens called the gradient
index lens and they simulated it with
off axis incident so therefore in this
case we have common so if you look at
where a these are right ray traces of
this lens for now accept please accept
the fact that the Razr curve but what I
wanted to solve you here what is the
caustic that I want I mentioned in the
last lecture you can see that in the
bottom side of this diagram there a
parts the kind of collapse onto each
other so you see that there is a
relatively uniformly distributed up to
this point approximately over here they
begin to accumulate near one side so
what you would expect in this case if
you put a piece of paper here you would
expect very strong light intensity near
the bottom of the of the beam so that is
what we call a caustic and I think in
the in the examples that the guys will
show you in the demo you will also see
some caustics form in real life but this
is what I was trying to describe last
night okay so pepper and Sabac I don't
know if you guys are ready but yeah
okay help me
so this is a similar demo to the one
that we showed now now we're going to
focus on a different thing or the
aberrations that we've been learning so
so far we've seen two type of
aberrations
in the system that we can identify very
easily so but first let me describe the
components so can you go back here so
for these one here we start with a laser
as usual and now we know that we have a
collie mating system so used to
condition the light and make a nice
plain wave at the output here and then
this plane wave illuminates this lens
which is a cylindrical lens shown here
and then we are just looking at the race
going on okay and this is for a green
laser is a wavelength of 532 nanometers
so now let's think about what's
happening actually so we can reduce the
exposure okay so we can see at this
point how the Rays start converging but
it's a bit hard to see here but that
they don't do not focus in at I'd point
they actually focus at different points
and as we'll see in the software this is
basically the experimental version of
the spherical aberration that we can see
and and in this case I'm eliminating the
lens with an unexplained wave so
therefore we can see this condition now
let's try to look at the case of coma as
seen from the top so what I'm doing is
that I'm rotating the lens and then we
can see how the Rays start forming an
isometric pattern and let me try to get
this closer together okay so now for
these all faxes plane wave we see that
some of the race go straight through and
some of the Rays form these caustics
here that we we just mentioned George
mentioned in the clean in the slides if
we were to put
perpendicular to this this one that I
have here which is hard because we need
actually a camera in order to show you
that with very very tiny pixels because
the actual point produced or the focus
produced by these lens would be very
small but if we do that we would see the
coma shape that we described and we're
actually going to see that in the
simulation software that we're gonna
show later okay Lisa yep is it possible
to put a business card can you put the
back of a business card there you're
saying perpendicular yeah you don't see
anything George this is a cylindrical
lens this is a little yeah and you see
what it's saying
these are cylindrical ends and and and
even with a CCD it's very very small
right you would need something in order
to measure it accurately something that
it's called a hook or knife edge test
for example where you actually can
measure the the what it's called a point
spread function and we're going to learn
that in the next part of the course all
right so let's let me show you another
type of operation now imaging a
transparency in this case I just have a
white light source that is illuminating
this object which is just a transparency
with a some grid in this case is going
to be just the agreed that George was
showing that is regular grid and I have
a single lens here it's a spherical lens
by convex with a short focal length so
therefore is the curvature is pretty
high and we expect to have a lot of
aberrations but the one that we're gonna
see here you can decrease exposure we're
gonna see the evidence of distortion
aberration of distortion and as you can
see here let's just focus there okay you
see how the lines similar to the slide
that that Georgia showed the lines are
actually curved by these lens although
these are supposed to be straight lines
they're curved let me just magnify a
little bit
now if we were to replace these with
another lens so for example this one
that has a longer focal length the
radius of curvature is larger so it's
basically not as curved as this one and
we're trying to do the same now what I'm
going to do is I'm going to use this
lens to image my transparency
now we can see how the lines look much
more straight the distortion aberration
got reduced so what type of aberration
did you guys see it was pincushion or
viral aberration the one that you saw
before I'm gonna put it again what do
you guys think
which one is this
I can hear just press the button please
Jenny Christian yep yep we see there the
evidence of the magnification of the
lines is may be hard for you Singapore
to see but here the lines basically get
curved to outside so in the break if you
guys want to come and just play with a
demo and change the focal length and see
where the images get formed and also
this one here you're welcome to do so
now we're going to switch to a numerical
software to show how we actually deal
with these in practice to optimize an
imaging system okay so people before we
move on to next topic which is green
media and Hamiltonian optics we want to
show how we actually period or design
the actual imaging optical system in
practice because you know we so far we
learn a lot of stuff you know the
Snell's law and the reflection and
refraction and the lensman cross flow
and peroxy optics and blah blah blah but
in real life we usually use a computer
software or so just to give you an
example you know since most of your
mechanical engines I'm like including
myself so let's think about this so if
you want to build some toy or we can
come out for your bike or car or toy and
what you what do you do is you start
with sketching write by hand submit the
conceptual model and then you go to
compute computer use the CAD program so
you peel the Trudy or 3d model and if
you have multiple element you can even
assemble it right in SolidWorks and you
know you can export a 3d model and if
your if your system is complicated
enough and you can also use finite
element model to analyze
strain or the no temperature variations
of blah blah blah so the same idea you
start with the conceptual model by
throwing few ways and then you apply a
lens low and you know the vulnerable
object distance plus one about image
distance should be 1 over F and some
time you want to use the waitress from
ethics but after that you go to computer
and use probably one of these so
commercialized software which is a code
five so it's not V is Roman number it's
called five and or zmax and auto
so the the code five and the next day
are in the industrial standards so and
they are pretty much the same so if you
if you can handle one of these then it's
very easy to do easy to use the other
one
so essentially what it does is it draws
some rays for you but very fast and
automatic eyes automated fashion because
you know if you have a cat for cat
program you can use actually to do this
ray tracing right so you just read two
away and whenever you meet the interface
you complete the incident angle to the
surface normal and apply the Snell's law
and completely refraction angle and do
the same thing right but it's very
complicated especially you have a 3d
model so this problem does there for you
and it has actually has very different
method to do ray tracing so first one is
just you know actually what we learn the
weight transfer matrices so it was a
rate regime based on the peroxy region
and also but probably it is one of the
reason that we use the software so it
does to relive everything just that in
situ squad it just tries to everywhere
in know based on david's natural and and
you can use it and actually it also has
a many built-in features to analyze
performance of optical system like
aberration so we
learn about the five five geometric
revelation you can analyze it and also
points real function or spot dire of it
which means if you have ideal imaging
system and let's say we had pouring
object and you should the images should
be also infinitely small point wise
cause you know it did they are conjugate
but in real life this is it never be
like this because you're you always have
the finite size of your finite size of
five finite size of focusing it so point
a function or spot diagram tells you how
big is your focus at the image plane and
it also analyze MTF or OTF it is
frequency analysis we'll cover later of
the course and probably the answer the
one of the another reason that we have
to use this obviously optimize the
optical system I I prepare some example
so we see
so this is the so we are using the zmax
no just one of the industry's standard
and if you open the GTX and this we have
at least cables it's basically a
landscaper data editor so filling in
this table means you specify your ticket
system I'll I'll describe how to do it
well also you have this once you fill in
this table then you can analyze a very
different stuff so for example it shows
the actual layout of the system like
this one this 2d layout and it even make
the authority model for you like this
one so you can check the actual shape of
the system and so this is it is called a
wafer so actually it is the horizontal
axis is actually where our pupil plane
and vertical axis is deviation of the
way at the image plane so if you have so
if your system is aberration free then
this line should be just hardened here
because all the ways meet at a point but
you know this is spectral system so you
have the deviation which means you have
aberration this guy's our spot diagram
so if you have point object in your
object space then you would probably
probably get this kind of focus that
your image image plane and so and this
is the same thing but for different
different wavelengths cause you know the
class is dispersive which means the
reflective of in reflective refractive
index is also function of wavelength so
if you change the wavelengths then this
spot at a table that is spot also change
so it analyzed this a dispersive
properly
okay so let me explain how to improve
Auto feeding the dogs table so let's say
we have this tool and system so what you
do is you you just the composer's your
system to multiple stories so for so for
example here I have the object so this
is by default a surface 0 and the first
surface should be the front surface of
the first lens here and the surface to
it is the back surface of the first
first lens and the surface 3 is this
front surface of second end and so on so
forth so in the table you have we have
the different multiple walls so each
role we present the one surface and the
radius means is the radius curvature of
the circuit so for example here since if
the object is flat so the radius
coverage infinity and thickness is the
distance from the surface to the next
surface so in this case T naught is the
distance from here to the first length
and in some case if you deal with the
infinite conjugates like telescope the
indeed is T naught could be infinity
right because your optics ad continued
and at the surface one I have the lens
which has the radius capital R 1 so I
have R 1 and thickness T 1 represent the
thickness of actual lens from here to
here and since it'll and after the
surface 1 I have the class so I just can
specify the type of glass which is PK 7
years in most more most common optical
glass and you just keep repeating this
one so once you have this and you have
the are you just you just fill in the
table the digimax and there are also
other parameter you have to specify so
as I said the wavelength and you also
need to choose the field parameter which
means basically this means of the
information so if your object is in
finite distance
then figure the means you'll feel the
means object height or size of your
object or if it's in infinity and you
can change that you can set the incident
angle to the system just like the
telescope so so for example I prepared
is accurate example from the lecture 3
so we had the two lens
both of them have the focal length of 10
here and here and if you went through
the deviation with at the several
different ways to derive but for the the
image distance actually it's not image
difference so image is located at Tory
Marinero behind the second lens and the
lateral magnification was negative 4 and
also you if you did the weight transfer
matrices then effective focal length was
2003 she's about 7 and the front focal
lens and peppercorns was the 10 over 3
it's about 3 right so in Zenith I
already filled in the table so
so this is the how the system look like
I have object here and I have the image
here and if you analyze the system then
it keeps you the older specification
here so here the effective focal length
is 6 point I don't know you can see but
he's in fact epochal and say 6 point 9 3
to 7 7 so we have 2 good argument here
and the paper currency is about 3.05 so
about sweet three and the perhaps your
magnification is negative four point
eight so we have slightly bigger
magnification in real system and so on
so forth and also I put an aperture stop
here so you can compute the time
diameter in the size and polish onto
entrance pew pew and exit people as well
so back yes I think the reason for the
discrepancy is that the lenses have
finite thickness in your model yes so in
your model as you can see in the layout
this is not in length and I just put the
the brief numbers 2 into the radius of
the lens so I just put it 10.5 and the
thickness of the lens is one millimeter
in this case so it's not very wide in
length so it gives you slightly
different numbers okay and the next
example is actually the Hubble Space
Telescope so professor bhava status
already mentioned the last lecture about
it is telescope so actually the after
another realized after they launched
this telescope and find out the image is
not good as it's supposed to be and they
form a committee to investigate the
Worcester problem and they eventually
figure out and indeed fix it but let's
just start with the structure
so this is a typical category in a type
telescope we have two mirrors are two
mirrors so one is primary mirror which
is concave hyperboloid hyperboloid
mirror and we have the secondary mirror
which is convex
hyperboloid amuse and the size of the
mirror is the diameter is about two
point four meter the primary mirror and
the second one is about I guess 60
centimeters or so and the distance
between these two mirror was the four
point nine four point nine mirror so
it's a huge our telescope and after
extensive investigation the the problem
was the actually the shape of the
primary mirror is not same as the
intelligent so the overall shape I mean
the curvature of the mirror was slightly
different than the design and it turns
out that even though I'm actually the
diameter of the these primary lens is
2.4 meters but to a dredge the
difference between the design of the
actual Jay was only 4 micrometer
so it's very very small the physical
difference so so just give you an idea
so the black hair like the Asian I'm
allowing myself the world tire diameter
of the word big hair is about 100
micrometers so so 4 micrometer is very
very small distance compared to the
diameter wise but it generally
introduced the huge spherical aberration
which means so ideally you have the all
these in commemoration meet at one point
but it has a different you know extended
focal point so if the effect was to
catastrophic so the image of galaxy
it was loop will look like this but
after they fix it look like this
so anyone does anyone know how NASA fix
this problem
you know the telescope was already in
orbit so you cannot take it back to the
ground and you know to fabricate another
big mirror it's also need co2 resource
so how can you fix yeah so actually what
they did is you know is this huge
telescope and they had a focal point and
have different cameras like cause you
know you deal with different spectrums
like feasible or so IR or UV and what
they did is actually and all this camera
I share the same book as a Papa plane so
what they did they actually sacrifice an
auto camera and put another mirror
system here to introduce introduce the
spherical aberration but in opposite
opposite fashion so they kind of cancel
the spherical aberration so I prepare
the example in zenith
with the real parameter so this is the
intended design of the Hubble Space
Telescope and as you can see the surface
number two and certain number three they
are primary and secondary mirror and
it's a hyperboloid mirror so you have we
have here so connect coefficient and
this is the simulation result if you
have simulation result of image if you
have the letter F but due to the
manufacturing process this chronic
coefficient was not this one so actually
it was negative one point zero two three
to four so overall shape it doesn't
change them much but the imaging
analysis it looks very plausible so
that's why we had the very blurred image
of galaxy and that's why you need to
deal with all this aberration in with
this computer software because you know
you can analyze easily and get
compensated
okay so last example is optimization so
for example let's let's think about this
one I want to design on some optical
system which is three lengths three
lenses here so these two high stimulants
is made by made by crown glass and this
negative lens is actually a Flint glass
and if you are not designed this one
then you have many parameters you can
tune right so the radius coverage of
each length each length so R 1 R 2 R 3
i-4 and i-5 i6 and also the thickness of
the lens and for total distance between
the lenses so we have the another six
parameters here so you have total 12
parameters you can choose and
you know it's hard to find the best
Panamera right because you need some
specification to satisfy so what you do
is you can just run the optimization
routine here so first process first
procedure is select variable as I del T
squared you can tune the radios of
coverage of lanes and the thickness and
the distance between lanes but so if you
wanna choose the variable you can tune
and this communion you define the
married function or sometimes it is
called object function or cost function
basically is you just select what you
want to minimize for example you wanna
minimize some spare collaboration for
given of the point or maybe you can
minimize or a spot diagram like the size
of focus or different of field point and
also you can add some constraints like
your total I mean C square system should
be like this it cannot long on the right
channel longer than certain distance or
the thickness of the lens can be
constrained to to the manufacturing
process so you can add some constraint
and then what you do is compute the
value of Mary function with the
different combination of the parameters
and this is automated the I to achieve
our process so the image can do it for
you
so this is the layout of the system I
start with her just for a fair you
I just went to two then random value of
properties I mean the trade barriers
here so I we have a three year lens this
year and parameter with the remark they
are variable so we had a trade barrier
student and this is a result of the
revelation at the initial state and the
spot diagram to it off of course
so since so as the bottom left figure I
have did 41 instant angry so one two
three four different colors here so this
poor role in spot I went means state
corresponding to this different incident
angle and this five columns means I have
five different dip dip occurs so the
center one is focused and they are
slightly difficult in thrown or pair and
I also have the spot diagram with
different wavelength so this one is
about five hundred and an emitter and is
about 480 nanometer is against this one
is 550 or so if your so this is in very
function
I already defined the Mary function is
right use the default method function in
this case so there are many many
different parameter you can tune but the
essence is you know minimize some
aberration at the image plane
so I'm gonna run this optimization with
the auto-update so you're going to see
the actual shape all these analysis in
for everywhere every iteration okay so
we start with this situation and I run
like 50 cycles here so every iteration
is update the system and it actually
improve the performance of system yeah
so yeah so after 50 iteration this focus
is moderniser
focus are much nicer and it doesn't
change them much but actually this case
that it secures a difference so actually
it improved a lot okay so after this is
optimized so if this is what usually
optical engineers - in industry when you
wanna design some optical system but the
is this the best of our best design I
mean can you say this is this the best
design are probably now right cause you
know you just wanna talk
minimum it can be just global minion so
it kind of sensitive to initial cost
initial value or constrain or whatever
and also we have there are many many
issues and in practical issue you have
to consider for example so in this case
let's see this October length the last
length the radius curvature this length
is 60 point eight and this one and those
the second radius is negative eight six
sixty eight point three so is they are
very similar which means it looks like
convex lenss right but they are very
deep they are slightly different but
they are not same so if you're not
assemble
system then you need to figure out which
switches surfaces foreign service
efficient is to like the back surface
fight but it's very similar so it's hard
to figure out so sometimes you need to
consider many many practical issues
cause in this case it's not good it's
not a good design because they are so
similar you better have the player come
back to is to figure out feature
services from and it's obvious facts of
this right so the next door step for you
I mean the optimization but you need to
interpret it interpret the reserve and
you need to take on take into account
many practical issues so basically which
means it means nothing right so that's
why we learn many many different I mean
Dermott collective you need to interpret
it and you need the experience and
insight okay that's what I prepare three
type of lenses eye health is my color
here very very interesting type of
lenses because the index of refraction
as opposed to normal lenses that you're
familiar with and we've been showing
over and over which is just a piece of
glass of a fixed index of refraction in
this case as we've been mentioning the
index of refraction changes as a
function in this case of radius so the
lens itself it looks like this rod here
so this is a 3d model just like a
cylinder of glass but its properties or
its optical properties especially
refractive index as you can see changes
and in this case commercially available
green lenses the way they're a
manufacturer that are constrained to to
be with a parabolic
index of refraction variation so at the
center so this is basically the distance
from the radial distances at the center
of the lens optical axis and this as we
propagate one as we move to one of the
extremes of the lens and we see that we
have the highest in this index of
refraction at the center and then the
case parabolically
to the sides and if you can notice here
the variation from top to bottom is
maybe hard to read but he
very very small so it goes from
something like one point six to one
point five nine or something like that
so it's very very small the index
variation however that is enough to
guide the light and cause that the race
to bend so one thing that we're gonna
see here is that Snell's law is
basically applied in a little bit
different way before we're interested in
curved surfaces and a ray hitting a
giving curved surface will Bend
accordingly to Snell's law and the index
of refraction in this case you see that
the faces of the lens to both sides are
flat so if the index of these was
basically uniform the plane wave would
just go straight through like a piece of
glass like in a window right but the
fact that this has a parabolic index of
refraction then it makes this race to
smoothly be guided and bent towards a
focus in this case on axis or off axis
is a 10-degree of axes playing wave
incidence
so what is the nice thing about these
things is that instead of only using you
can think about the refractive power of
these two surfaces these type of lens
uses the entire volume of the lens to
guide the race so therefore the actual
if I put a screen here and see the
intersections of all these rays in that
screen this is the spot diagram that
we've been seeing also in the other ones
we see that for the on axis point we
have a tied focus right which typically
is smaller this in this case the focal
length the effective focal length of the
system is is very small so if you can if
you try to do this with a normal
refractive lens with curved served
surface services you would need to have
a very sort of like really really curved
lens and as we know from the demos and
from the discussions those type of
lenses will be really underrated
however these even though that has a
short focal length it still focuses into
a very tight focus because it uses as I
said the entire volume now here this is
an example of coma so if I actually show
I can even focus a little bit in into
how the Rays are coming and we see the
similar caustics that you saw in the
demo
and in the in the slides before but if
we put a screen here this is the actual
shape of the race intersecting that
plane so these two are circles of
completion and just just to give you a
preview this is basically for the on
axis point so this is what geometrical
optics predict the bottom one right so
the geometrical optics predict just how
rays intersect a given plane it doesn't
take into account diffraction that we're
going to learn in the next part of the
course
however diffraction is very important in
an optical system because it regulates
resolution and many of you have been
asking about numerical aperture and how
that depends on resolution and what
happens changing the aperture stop and
so on and so forth well these is also
very related to these points for
function that you can see here and as as
you can see it's a bright spot and then
there's concentric rings which is the
the way you can see the diffraction
effect of that point so this is a
geometrical version this is the
diffraction version or the more
realistic version of how light would
defragged and in order to compute these
will will basically need the math and
the tools that we're going to learn the
second part of the course but this is
just giving you the type of designs and
analysis that we could do in this
software so any questions before we're
gonna break right George or yeah yeah
any questions before you break
I think it's time to continue
and before I move on are there any
questions about I don't know about
anything about the homeworks about the
lectures about imaging about focusing
Jupiter Langston I guess I don't believe
in quizzes actually I hate given exams
but I have to produce grades that
educators asked for grades to the except
they get established way is with exams
but anyway and I also had two reviews
because of the same reason because if
you suggest somehow that the quiz is
important of course the quiz is not
important except for the Registrar the
only significant that perhaps exams have
is that they really force you to study
right I mean if you didn't have an exam
then you say well maybe I'll study later
maybe I'll study in the summer you know
when I go out waterskiing I can study
optics while they a water-ski but but
anyway so I try to when I in my classes
I try to the emphasize exams as much as
possible and the assumption is that if
throughout the course of the class you
have learned the material then you will
do well in the exam whereas to do the
opposite to try to satisfy the needs of
the exam is very limited because then
what you get out of it you know you
waste good time so so that's why I try
to not mention exams as far as I can
help it as I not mention but I know I've
been a student too so at the time I
would also be very worried about exam
well anyway so this is the time is there
any any sort of whatever well if you
questions or
okay
bless I'm going to come to from three
down right three how to do in the
auctions why okay you can still
interrupt or also we have the class
forums I mean the online chatroom or
whatever it's called the forum of the
column so of course you can post
questions there and we'll be diligently
checking also sent an announcement by
email you may have not seen it yet but
we've posted some solved problems
actually the problems have been posted
since Monday but so the problems are
solved so you don't really have to so
this is the homework for next week you
don't have to actually turn in the
homework on the quiz date that would
have been cruel but we actually get you
some solved problems so you can go over
them and of course the the
recommendation is to to really pretend
that you don't have the solution they
have to solve it yourself and then check
it check your solution against the hours
and of course you don't have to turn in
your solution okay
and that's I'll stop talking about the
exam now unless someone has a question
now or later
okay so Pepe already gave an
introduction to the topic of gradient
index or green optics and he mentioned
that the motivation is basically that if
you can focus the light by using a non
non uniform material the material whose
index of refraction is variable
perhaps you can get away by also using a
material who surface is flat so you
don't have to use these awkward
spherical shapes that we know from
everyday optics maybe you can have
something that looks like a slab or or
or or a plate and nevertheless this flat
looking plate would might have the
ability to focus life so the way you can
do first of all can you do this kind of
thing can you make an element
with variable index of refraction so
there's two well as many ways but in in
industry in manufacturing practices
there's two ways that have been good and
I will describe them one is called the
ion exchange and what they do is they
take a glass rod which is doped with a
certain kind of ions typically they use
a sodium in the glass rod then I have a
glass melt at a very high temperature of
course with a different kind of iron
typically lithium and what happens is
they slowly dip the glass rod into the
glass melt normally period of several
hours what happens is the little Mayans
they kick out some of the sodium ions
from the glass rods they basically
penetrate the rod and they kick out some
of the sodium ions but of course the
index of refraction depends on what kind
of ions you have inside the glass so and
also because the ions they propagate at
a finite speed what will happen here is
in the exterior of the glass rod you get
a lot of ion exchange so the index
changes a lot but near the center if you
get your time in right near the center
you don't get any exchange at all so you
basically have in the center that you
only have sodomite ions near the
exterior both sodium and lithium so you
can get a situation where the index of
refraction is different in the center
than it is at the at the edge and then
what they do of course is after the fins
they finish the process they chop this
glass rod into slabs and then each one
of those slabs is begins to become a
green lens and as I said before they do
it typically so that the index is higher
in the center and lower at the edges so
this kind of thing I will do in detail
later but it focuses if you look at it
in cross-section it focuses like so if
you illuminate it with with parallel
bundle over
you can think of the ethics of
refraction as an attraction of sorts for
leis so they're a the tempers at the
lower index it gets attracted by the
high index it bends and as a result by
the time their eyes come out they
actually come out as a focusing bundle
so you can use this element to to focus
light so it's just like a lens but it is
a flat self and because the ion exchange
is actually diffusion driven I don't
want to go into the mathematics of
diffusion but it turns out that that is
you get a specific form of the index of
refraction that you get in this case is
quadratic or parabolic so it is maximum
in the center and then it drops as the
square as a radius square where R is the
radius of the radius in the in the slab
times some coefficient alpha and I will
show in a second that this coefficient
alpha and the thickness of the lens the
product of the two actually specifies
the focal length so this looks like a
funny formula but if you think about it
alpha must have dimensions that are
inverse distance square okay so that the
product alpha times R square is
non-dimensional and B is distance so
therefore the units here are a distance
so indeed the quantity over there is a
distance if this is the focal length
okay so let's see that why this is the
case so in again we've done it several
times so you must be tired of it by now
I just lost my memory the other things
also must not forget okay so so in
typical fashion we will use a thermos
principle in order to calculate what's
happening here
so Farabaugh says that if you since
here is my element okay
and I have two days but both started at
infinity right so the Rays share a very
origin the starting point and then my
desire here my intended outcome is that
these two rays meet because I want the
element to focus the life so firm a such
that the optical path length here on
axis and the optical path length out
here all foxes the must be equal because
the range started at the same point they
end up at the same point the optical
path length must be equal so what is the
optical path length so this derivation
here is very highly hand-waving and very
very inaccurate but bear with me in the
in the next half hour I will tell you
how this calculation can be done exactly
but for now this is kind of an
interesting way to think about the green
so will do it anyway
it's also in the book so you don't
really have to copy my derivation here
you already have it to do in two places
one is in the book and the other is the
note so maybe you can just look at it
and don't bother to write it up okay
so let's compare the two let's compare
the horn access today which basically
goes straight okay this is the optical
axis hmm and there are facts is life
which as pepper already mentioned or
facts today will actually do something
like this it will Bend and then of
course after it comes out in phase space
it will it will go straight you know
that for sure but in fact the gradient
in the X medium that I will bend it will
follow a continues the continues kind of
a curved trajectory this is also
counterintuitive because in our
experience light rays go straight but
such is life you know physics sometimes
have an expected outcomes if you put a
ray in the gradient index medium indeed
it will it will bend onto a curved
trajectory okay so let's put some
notation here so we call this our this
is the
use the radial direction we'll call this
D this is the thickness of the grain
we'll call this F this is the focal
length okay and for the sake of of this
derivation you know let me let me let me
actually blow it and I will redraw it
forgive me I just want to emphasize this
curvature okay so will actually neglect
this distance this space over there I
will pretend it's not there and this is
consistent with the per actual
approximation okay here is our here is d
again and here is heaven that's the
optical axis okay so so what let's do
the first of all the optical path length
for the on axis ray well the optical
path length equals the index find the
distance integrated over the part of the
right so this would equal the index at
radius equals zero that is the index
over here at the center of the lens
times the thickness of the lens plus the
focal length right of course outside the
lens index we assumed to be one you
could have an imagined line but let's
not do that now so so this is where it
comes from and yeah according to from
which principle it says that the rate is
in a such a way that it tries to
minimize the path but so why does the
Ray goes from lower index to higher
index means if it goes into higher index
material actually it is increasing its
optical path length but it will Bend
that's what Snell says it will Bend so
that way our total path length actually
decreases
I think well I don't know but maybe the
answer to his question is that although
I think what he says is that it will
actually travel through a longer optical
path in the medium because it goes into
the bigger index but I think that that's
kept that's counteracted by the extra
that it would be outside isn't it
in the air I think yeah because it will
have to go through this hypotenuse over
here it will actually yes
so the overall path is actually
minimized right and this is the only way
that can be minimized yeah okay okay so
what behold then let's do what you just
said so what about the Ray that goes
that is arrived at a higher elevation so
here I'm going to treat big time I'm
going to pretend that all of this the
right bands of course right but I'm
going to pretend that throughout all
this travel here there is the same index
of refraction okay this is a gross
approximation but I will do it so
therefore this is so and not another
name for it is and it is also equal to n
max the optical part for N 1 will be
equal n max 1 minus alpha R square over
2 which is the electric refraction at
the elevated position and then as you
pointed out it will go through this
increased distance which is the
hypotenuse of the of the triangle so it
is really R square plus F squared so
again I pretended that this R and this R
are the same right I won't emphasize
that this is an approximation and
untreated
okay the next is actually easy from the
one all I'm going to do and the division
I'm going to do is to remind you I'm
going to do the paraxial approximation
which basically says that R is much less
than F that means that square root of
the square root can be written as pull
the F outside 1 plus R square over F
square approximately equal F 1 plus R
square over 2 F square and this is
actually equal to F plus R square over 2
F ok so now let's write down firmly in
support the math says that o PL 0 must
be equal to o PL 1 so a PL 0 was n max
times D plus F and OPL 1 was again if
you look at it from this equation or PL
one was this expression times D of
course I forgot to put a dick here times
this distance so it is equal to n max 1
minus alpha R square over 2 times D plus
F plus whatever R square over 2 F ok now
a number of things will cancel this will
eat this one this will eat this one and
all that is left is that n max times
alpha R square over 2 D with a minus
sign plus R square over 2 F equals zero
and now we can say that the quadratic
dependence for the index of refraction
that
Chow's is convenient because the same
quadratic came out of the paraxial
approximation of the square so I can
just cancel the two quadratics and the
twos also cancel and I get that F equals
the inverse of all of that 1 over N max
times alpha times D this is how we find
the focal length of the green so so this
disappears a little bit mysterious what
happened here how did they pull this out
of the Hat well what it really means the
fact that his this has a solution means
that this element actually succeeds in
foucaultian what I mean is the following
suppose that what came out of this
equation was not the - quadratic that
canceled but suppose I got something
like this suppose I got minus n max
alpha R a cube for example over 2 times
D plus R square over 2 F equals 0 well
if this is what I had gotten amount of
luck this is not a proper length so what
I'm trying to say is suppose that I made
the green with a cubic dependence or the
index of refraction rather quadratic
then this is the question I would've
arrived instead now I cannot cancel our
anymore so it means that each array goes
to a different focal point
so therefore this element does not focus
okay so this would not be this would not
be a good lens actually the cubic
dependence has some other interesting
properties but but these are perhaps for
a more advanced for advanced class
nevertheless the quadratic dependence is
very convenient because as you can see
it drops out of here it means that have
a unique focal length at which all the
Rays focus at least in the paraxial
approximation okay so that is the story
of the quadratic grin how comes about
and how how derive its focal length and
I would also say that even though my
derivation here was cheating
basically the way I'd see today's I
assumed that be the thickness of the
lens is also very small so it is very
similar to the thin lens approximation
there nevertheless this equation is
actually true even for the thicker
lenses so that's kind of innocent okay
the second way people make green optics
is actually a combination of melding and
grinding and the way they do that
results in a different profile that now
the index of refraction vial is along
the longitudinal the axial dimension so
what does this mean so what to do first
is they start with a stack of glasses of
different indices of refraction so here
and in the rest of this lecture the sort
of the darkness indicates index of
refraction so gray a dark grey means
high index light grey means low index so
they stack them then the dual they meld
them which basically means that they
mold and melt okay so after they melt
them basically the index of refraction
the variation becomes kind of continuous
because as they melt the different
indices will kind of diffuse into each
other so you will get a quasi continuous
variation alembics and the final thing
is they do is they actually polish this
stack that they got with a continuous
index variation they polish it so now
the thing actually is the properties
first of all it is a lens in the
traditional sense because it is it has a
spherical surface so it focuses by
virtue of refraction but in addition it
has an actual index profile so here the
motivation is actually a little bit
different than what pepper and I
described earlier
so clearly did not end up with a slab
looking element I still have a curvature
but the motivation is different
it's a little bit similar to in the
previous lecture I think we mentioned
something called the Schmidt collection
or the Smith telescope there's a very
similar motivation if you take a uniform
index sphere okay we learned we saw this
a couple of times that in general this
produces an elaborated elaborated image
even if it is oriented properly this is
supposed to be a plane or convex lenss
oriented in the correct way that is the
plane wave goes to the spherical surface
but still this does not completely
cancel vibration there is still some
residue so what you do then with if you
modify the index of refraction in the
interior of the surface what happens is
this way let's take the paraxial ray
over here
it hits at a relatively high index where
is this ray it hits at a relatively
lower index what this means if you want
to look at it from the geometrical
optics point of view there's also a wave
optics point of view that is interesting
but from the geometrical objects point
of view means that the Snell's law of
refraction here will be milder for this
ray than for this ray
therefore this ray which used to focus
kind of violently and produce a
spherical aberration here now it will be
pulled out words and therefore it will
be forced to go through the proper focal
point over here as the paraxial rays so
this is a clever way to fix spiracle
aberration and I don't know who invented
this first I believe and maybe colonies
and collectibles I believe it'll go
Duncan Moore at the Rochester yeah who
came up with this this particular at
least I've seen a paper of his which is
old as I am 1971 that discusses this
optimization okay
so another two sort of the basic
principles of grains I would like to
spend a little bit of time I don't think
we'll finish today will probably have to
Lake into next Wednesday after the quiz
but I'll get us started on how do we
properly compute things like index I
mean like focal length for a gradient
index traction so would like to ask the
question more generally if I have a
medium whose index of refraction varies
as a function of position so here this
cloud is meant to indicate a variable
index so we have high index low index in
different position so this is like a
generalized grin if you wish we saw two
types
you saw quadratic and Gasol axial but it
could be really in principle I should be
able to fabricate something like this
not that it is readily doable but let's
consider it for the sake of our you okay
so you see this diagram before in fact I
copied this from an earlier slide one of
the previews I think during this first
lecture or second lecture so we don't
know really what path array will follow
through this variable index menu but you
know one thing we know that if you
integrate this quantity if you integrate
the index of refraction times the length
the you know the elemental length along
the part of the array we know that
Fermont requires that this quantity
should be minimized okay so how do we
deal with that I mean what can we do
about this now some of you who have
taken advanced mechanics classes this
sounds a little bit familiar
what is the name that comes to mind when
you see this kind of expression so let
me repeat again every situation where
something in this case a light ray but
it could be something else
a particle or some some
an allergy can follow a number of
different trajectories it can go this
way or you can go this way or this way
and so on to give you an average sort of
trivial example if I drop my marker here
it goes down straight right so you may
have wonder well why does go does it go
down straight what does it not do
something like this or like this or
whatever why does it not go up for that
matter will it go straight because well
because gravity pulls it down right so
well we're very happy with that we know
from experience that things fall
straight but we also know that gravity
does not always produce straight
trajectories for example the earth
thankfully there follows a circular or
actually elliptical trajectory around
the Sun that's very fortunate because if
the earth had a way to escape from the
Sun then we would all be there would not
even exist so so another theory is that
you know gravity can produce straight
ejector can produce elliptical
trajectories are hyper below and
hyperbolic alpha actors and so on and so
forth so you know there's more to it
than the straight falling of the of the
of the Apple in Newton schedule of the
market makers so anyway so there is I'm
giving all the scramble is because this
is this may be a familiar view from the
field of mechanics and the buzz word
that comes to mind when you have this
kind of integral with something inside
then you demand that the integral itself
is maximized this Lagrangian it is like
the Lagrangian principle that you may
have seen and I think those of you who -
- w3 saw it did you or did you not
actually I don't know
is anybody in a class who took to the
blow three you should because to w4z
prerequisites in the principal if you
were here it means you've taken both to
double your three into the below for
with a lost dollar and a graduate or or
they lost their voice order to sleep
anyway
or there's a no possibility that would
not remember if we were total lagrangian
sin to w3f anyway but i will not do it
this way so so in turns out i can solve
this problem using a lagrangian
formulation but i will not do it this
way did someone want to speaker i show
the camera movement what simon about the
speaker I was just going to say yes we
did have thank you well even if you did
I I guess I will not do it with the
lagrangians because well the math is too
complicated and would scan over for the
detour here nevertheless I will use a
mechanical analogy so for a moment we'll
take a break form optics and I guess to
justify why to double afford is a
prerequisite for this class I will show
this and I will turn against the next 15
or so minutes to establish the
connection between this mechanical
system and light rays for now let's
forget about light rays and giving you a
it's actually particle so it looks like
a car but it's a really small so don't
ever have to worry about moments of
inertia and all of that stuff there's
also frictionless so I don't have any
dissipation at all in the system this
particle is attached to a rigid to a in
immobile rigid wall with a spring and
basically what I can do is I can inject
some energy into the particle by it by
pulling it straight so if I do that then
we know that the system has two forms of
energy when I actually pull it what I do
is a star energy in the spring the
spring is not happy having its energy
stored it will want to give it away what
does it mean to give a wave energy it
means to converted to kinetic it means
the spring will start pulling the
particle back as the particle is pulled
back it will accelerate it will lead to
maximum velocity we know that from
classical mechanics approximately where
the spring has zero displacement and at
this point all of the energy has been
converted to kinetic and of course it
will not stop there because it is moving
and there's nothing to stop it it will
keep moving but it moves now it is
beginning to push to squeeze the spring
so now the particle is giving back its
energy into the spring until all of the
energy has been given now the spring is
which doesn't like to be squeezed it
starts giving the energy back to the
particle and of course on and on they go
this will execute an oscillating motion
forever because I neglected all
dissipation and of course what is
happening in this case is that at any
given moment the particle will have a
mix and in the system will have a mix of
kinetic energy and energy stored into
the spring which is referred to as
potential energy so of course the
kinetic energy is given by one-half
times the mass times the velocity
squared if you remember the definition
of momentum momentum equals mass times
velocity so I can also write it like
this
momentum square over two times the mass
and the potential energy actually the
potential energy can have very different
forms depending on what is exactly the
energy storage element in the case of a
spring actually in the case of a hook
spring a linear spring it is simply
equal to the square of the displacement
of the spring with respect to its rest
position so anyway no matter what these
things are
the Sun must be constant because again
energy cannot be gained the last great
system so the sum of the potential and
kinetic energy is always conserved okay
so that is one thing it is very good now
let me let me play a game so first of
all I rewrite the energy I will
basically repeat what is on the slide
that will repeat it on the on the
whiteboard here okay
so let me write the energy I will read
it in a funny way I will add it as a
function of the momentum P and the
position two and now in the previous
slide these were actually scalar
quantities because this thing moves in
only along a straight line but in
general of course I can have positions
and momenta that are vectors right so
okay so I switch to vector notation I
hope that doesn't confuse you probably
okay so let me write it down now so in
the vector case this will equal the
vector the momentum vector divided by 2
m plus 1/2 M by 1/2 K they the spring
constant times the position vector
square now what I'm going to do for fun
is I'm going to take derivatives of this
quantity with respect to the position
and the momentum of course since this
became vector now these are not exactly
derivatives they are gradients so
basically what I will do is I will
compute the gradient of H with respect
to P okay this also this is a matter of
notation I will denote it as the h DP
okay so what is the HDP well
it is what I have to do is I have to
compute clearly this term does not play
because it only depends on true
but this term contains the momentum so
if I do it to tool actually you can do
it it is actually P times it's game over
them that's it nothing else and then I'm
going to do the same with respect to
tool and I'm going to call it D H the
two again it is a gradient with respect
to Q and for the same reason the other
one it will turn out to be K times two
okay then again for fun I am NOT
justifying what I'm doing it's just come
in sort of some cabinet of some some
sort of Epiphany and we write down two
differential equations one of them and
relate this one it is already on your
screen there DQ DT what is that that's
the time derivative of the displacement
where for it is the velocity so we'll
set it to be equal to the gradient of
the quantity H of the energy that we
call we call that H for some standard
error with respect to the opposite
variable
this was the derivative of Q I will take
its gradient with respect to P okay and
let's see what I get this way I get P
over m no that's not correct actually
this equation is not particularly
insightful this is the definition of
momentum isn't it right if you recall
this means recall like in cartoons
there's a little you know cloud the
momentum is defined as M times V to DT
that's a definition of momentum so
whether by doing this trick by taking
the gradient of the energy with respect
to the momentum and setting it equal to
the velocity
I got a true result I got a result that
happens to be true now let's see if I
can pull the opposite trick if I can
take the derivative with respect to time
of the momentum that is that's like an
acceleration quantity right because
momentum is velocity so it's derivative
is explanation and see if I can get away
by putting it equal to the derivative of
this quantity H with respect to the
other variable in this case I cannot
quite get away with it I have to put a
minus sign and what do I get if I put a
minus sign k2 what's this no
kay - it is carry x displacement so what
are the units of Hookes constant times
displacement force what is on the
left-hand side okay I said acceleration
actually I should have corrected myself
and I should have said proportional to
acceleration this is the derivative of
the momentum so therefore it is force so
what is this then the question that I go
to it is the first balance on the it is
Newton's law actually on the particle it
says that the force pulled in the
perfectly it is the restoring force due
to the spring and what are the two
equations combined actually the Terkel
the equations combined they're what we
call the equations of motion the
question of motion by definition are
first order differential equations that
link the dynamical variables that is the
position the displacement and the
velocity so this is what they did they
got a set of differential equations that
linked the displacement to the momentum
that is the velocity and again the
derivative of the displacement back to
the answer the derivative of the
velocity the derivative of the momentum
back to the displacement itself so this
if you look at it if you meet this
middle part if you look at it like this
these are simply as I said these are the
equations of motion if you look at this
part over here this is of course a more
general result that's why I bother to do
it and this is where the connection to
optics will come from next week
this set of course the set of equations
is called Hamiltonian and I will put in
brackets canonical I don't particularly
like this theorem but anyway some people
use it
Hamiltonian canonical the questions so
the point at which I will say just this
and then I will stop and continue next
time the point is that this is now a
much more general property even though I
sold that it is true for the specific
case of a mass spring system without the
damper I saw that to be true but this is
a more general property for any
dynamical system that has a conserved
quantity so in the case of the
mechanical system the conserved quantity
is energy if this conserved quantity can
be written as a function of the
dynamical variables in this case of the
mechanical system position and momentum
okay so if this were I started a long
sentence let me close my sentence so if
if a is a dynamical system has a
conserved quantity then I can write down
a set of Hamiltonian equations for this
dynamical system we times of the
gradients of this conserved quantity and
that will be the equations of motion
okay so what we will do then next time
is we will actually solve in almost
one-to-one analogy that on by the way
this conserved quantity that will call
the energy in the case of the mechanical
system in general it's called the
Hamiltonian so H itself is the
Hamiltonian so what we'll do next is
will actually show how can derive a
Hamiltonian for a for for an optical
system and of course when we write
equations of motion Falaise will get
there a trajectories okay therefore we
will be able to retrace the reason this
is interesting because the potential
actually will turn out to be the index
of refraction and in the same way that
the potential here if you go back to my
equation
the potential turned out to be a
function of position in the gradient
index optics that we mentioned just now
the index of refraction is a function of
position so we will see that in the case
of the green element the index of
refraction actually has the role of the
potential and it is this potential that
draws they're very similar to gravity
the potential that draws the Sun I mean
the draws the earth to rotate around the
Sun and not the hardware on the same way
the potential draws the earth to rotate
around the Sun it is a similar or should
say analogous kind of potential that
grows raised in words in a green element
and causes them to focus we will pick up
on that next time and you know obviously
don't worry about this for the quiz
right so I'm not crazy so yeah so will
will be properly conclude this topic
next time it will be next week's over