Lec 24 – Lens Aberrations


welcome you all to this course on electron
diffraction and imaging ok in todays class we will discuss various lens aberrations and
its effect or the resolution of the microscope ok the first question which arises is that
ah why do we use lenses because we know that we use lenses to magnify objects why do we
have to magnify objects because our eye has got ah limit on the extent to which the objects
can be resolved if we wanted to see features which are smaller than the resolution power
of the eye then we have to magnify the image what is the need for magnifying the image
the reason essentially is that though the light if we use for example if we use a light
as a probe then whats going to happen is that or the ultimate resolution is given by the
wavelength of the radiation lambda ok so that means that in the case of light generally
it is about something like five hundred nanometers is going to be there wavelength of the radiation
that means that there are features which are of the order of five hundred nanometers is
resolved whenever light falls on the material ok even in this room if you look at it when
the light is falling onto this table the features of this order are essentially resolved but
our eye is not able to see it because the resolving power of our eye is about something
like roughly zero point one millimeter to overcome this we have to magnify that features
so that that separation between them becomes greater than the resolving power of the eye
that is why we require a lens to magnify it but when we use a lens ok especially what
all the features which we look for the lens what all the characteristics which the lens
should have one of the characteristics of the lens is that you should do a faithful
reproduction what is the reality of the system the simplest case which we can take of is
a magnifying lens which we can which we normally use when we cant read objects properly but
this is an example which is being given here with the lens we are trying to magnify objects
this side you can see the normal image and here its the magnified image which we can
see but if we look at it we can immediately make out that the image is curved it is not
in a straight line ok the magnification appears to be different from edge to the center and
the center it is focused well for example in this case and the edge it is out of focus
all this problems occur because a simple lens which we use it to magnify has got of lot
of aberrations which are associated with it because of this aberration we are not able
to get a faithful reproduction of the object in the image in fact ah one point which i
want to quote at this the statements which have been made by maxwell in eighteen fifty
eight he has talked about what is the properties of an ideal image one each point in the object
ok there should be an equivalent point in the image when we form an image using a microscope
it could be a optical microscope or it could be a ah electron microscope and then when
we have an object ok this is what essentially is given here
this is an ah for example here if you consider a point object along the optic axis for an
ideal lens a point image should be formed so if the resolving power of the lens like
for example in the case of an electron microscope which is operating at it two hundred k e v
the wavelength of the radiation lambda is of the order of zero point zero zero two five
nanometer that means this is the limit of resolution so for an ideal lens all the atoms
in the crystal should be resolved but thats not what it happens this is because of various
lens aberrations which are associated with the lens which we will come to shortly
the next point which he wants to ah which he has mentioned is that object and the image
are geometrically similar ok what does it mean that is if [ob/object]object has got
a particular shape the same shape should be reproduced in the image as well and the third
is if the object is planar and perpendicular to the optic axis so is the image these two
points are shown in this slide here if you see it this is a b c is an object this is
a lens system which magnifies it ok we are able to get a magnified image on the screen
if we measure the ratio between e a b here and a b here b c here and b c here ok and
also the angle between them they are faithfully maintained then only we can say that this
is a true [repru/reproduction] reproduction of the image what normally happens in many
of the lenses one for an example which is shown here is as square grid is taken as the
object and you find that the grid itself is distorted this is the way it appears when
this appears we call its a pin cushion distortion and this is a another way in which a square
grid can appear then we call it as a barrel distortion here there is a rotation is also
associated with it then its called as a spiral distortion then there is an another distortion
which can come is that the image is not formed perpendicular to the optic axis for an object
which is there like in this case is for a plano convex lens you can see that that image
is formed in a curved space ok so this is also an another type of an aberration so we
have seen curvature then different types of distortions and there is an another aberration
which comes because of the curvature of the lens the spherical lens this aberration is
normally called as the spherical aberration before we go into understand the aberration
ok what we should know is that why does this aberrations arise in the lens in the first
place in tenth or ah eleventh standard everybody has studied about a geometrical optics and
in this we consider that if a lens is there and if we have an object which is kept along
that optic axis and then we know that the rays which are optic axis go through the focus
and the ray which passes through the center of the lens the place where they meet this
is where the image comes this is the object this is the image if this distance is u and
this distance is v and if it is f in a one by u plus one by v equals one by f this is
the formula which we use to construct that image but what one should understand is that
when this image is being constructed ok there is one condition which is there we call that
the rays are paraxial rays what does paraxial rays mean when the light from the one medium
which is air enters into the lens ok since the refractive index is different it has to
obey the lens na the snells law ok what is snells law snells law says that sin theta
by sin phi equals mu the refractive index that is if this is the normal this is theta
and this angle is phi ok this snells has to be obeyed and this is the rule which governs
how the rays are bend when it enters from one medium to an another medium if we apply
this rule ok then what essentially will happen will for an ah ray from an object which is
parallel to axis but not very close to it will it obey this rule or not ok this law
this formula itself this gaussian formula itself is derived ok for the condition that
these are all paraxial rays what are paraxial rays paraxial rays are the rays for which
this value of the theta and phi becomes so small that this mu can be written as theta
by phi that is sin theta can be approximated to theta and sin phi can be approximated to
phi but what is normally happening is that the actual expansion for sin theta is sin
theta equals theta minus theta cube by three factorial plus theta to the power of five
by five factorial minus theta to the power of seven by seven factorial this is the way
the sin has to be expanded so only when the value of theta is very small
that higher order tempts can be neglected and sin theta can be approximated that means
that for rays which are very close to the optic axis only this rule is valid and this
gaussian formula is valid but that is not the case when the rays are coming ok falling
on the lens ah on all points in such a case we have to use formula or the approximation
which we can use it is this is the approximation sin theta can be if we use this approximation
till which is closes to the reality then whats going to happen is that the rays which are
coming closer to the optic axis they form image at one particular point the ray which
is coming at the edge of the lens ok they are focused to a point in all the cases the
snells has been used to ah trace the ah ray which emanates from the object and forms an
image ok in this so now we can make out that ah the ray which is ah further away from the
optic axis they form an image on the gaussian plane because this plane ah where the image
is formed for paraxial rays this called as the gaussian plane in this the image size
is very large ok here that image size is again so in between there is a region where that
image size is minimum this is called as the disc of these confusion which you have all
studied ah so this ah same rate racing itself we can
look at it in a different way instead of a ray if we consider the light as a way which
is propagating ok then we can think of ah from the object from a particular point in
that object on that optic axis the spherical waves are emanated ah emanating and these
waves one can make out that they come and meet at that lens what does that lens do the
lens essentially gives a phase shift ok so that the crustal has changed to this particular
shape and when this propagates in these direction it will come back to a focus at a particular
point which is that image point so here it is only an advancing the phase shift which
is doing it which is varying as we go away from the optic axis in the case of a convex
lens if we consider it here whats going to happen it is essentially delaying that phase
which is happening so in this case also for a point object we are able to form a point
image and this lenses are considered as a ideal lenses why i am mentioning this point
is that later when we talk about high resolution microscopy the ah we have to consider lens
as phase shifters let us come back to the spherical aberration what is spherical aberration
the spherical aberration is is nothing but when we try to draw the ray diagram for a
lens ok if we gives the case of paraxial ray this is the formula which we will be using
to find out the position of that object and the position of that image but when the lenses
of finite dimension and the angle which submits with the ah ray surface is not very small
in such cases we have to use this particular formula ok and if we use that then the ah
it gives rise to a situation where the lens has different focal lengths depending upon
which point the ray is entering the lens at what or that is what distance from the ah
center of the lens the ray is entering the lens this spherical aberration ah what is
its effect which we can calculate it because what we find here is that its not one focal
length ok that is ah for a paraxial ray this is the focal point and we can assume that
ah f is the focal length and the ray which is coming at the edge of that sample that
is focused at this particular point so this delta f is the variation in focal length which
is occurring this delta f can be represented in terms of some constant to ah the distance
from the center to the ah edge or its the radius of the lens that is ah into x square
plus c four into x to the power of plus per higher order terms it can be written
but using simple geometry we can calculate that this x itself equals if this distance
is f one and into tan alpha this is that angle which is submits here is alpha and then it
can written as ah assuming that delta f very small this can be written as f tan alpha and
also the assumption that the alpha is small similarly on the gaussian plane what is going
to be the spread of that image that r s is equal to delta f into tan alpha so this is
delta f into alpha we can substitute for delta f with respect to this is the only term which
is being used first time when we submit ah substitute this one the value of r s turns
out to be ok a factor c s into alpha to the power of three ok this is ah the so that is
ah the diameter of that image ok its going to be two c alpha c s into alpha to the power
of three and that the disc of least confusion the diameter is going to be zero point five
c s into alpha to the power of three ok so for a point object we can make out that its
not a point image there is a disk of least confusion is what we get it this is one aberration
ok this aberration itself we have ah just shown it ah for an [pa/point]point object
which is going there on the optic axis and we dont get point image but its an image which
is ah spread out in ah space this if we consider it as a waves which are emanating from the
sample ok and then ah propagating then we can make out that the rays which are away
from the optic axis they are bent ok the clusters has been bent a little bit more forward because
of that only this sort of aberration which is occurring so so far we have considered
a spherical aberration for a point object which is lying on the optic axis but normally
object have a finite size like in this example so the rays which are coming away from the
optic axis from this point on the sample what will happen to it ok this also leads to an
error or an aberration and this aberration is called as the comma it is like a coma the
shape of the aberration looks like that of a coma
here one can make out that the ray which is passing through the center of the lens that
goes on deviated ok and for this particular ray and the rays which are close to it paraxial
rays ok the focal point is here so its a small point at which we form that image and the
rays which are far away there is going to be a spread and that is what essentially is
being shown from here to here as we go that size of the image itself is becoming large
so this is like that of a coma which you can see this is that shape ok this aberration
also has to be so these are all the two aberrations of the microscope and this is also related
to the spherical aberration coefficient c s ok what is the other aberration which arises
in a microscope this is called as the astigmatism that is if the lens itself is not perfectly
spherical the surface is not spherical here we find that there are some distortions ok
then this will give rise to two different focal points in two perpendicular directions
that is if if a ray enters that lens in this one and the ray enters in these direction
it will be focused at two different points these gives rise to not a sharp image but
an image which is distorted how this can be corrected this can be corrected by putting
in front of this lens a non cylindrical lenses that non cylindrical lenses initially is a
lens like this one another is in a perpendicular lens like this this sort of lenses if we use
that we can use them to compensate and correct the astigmatism ok
how is it done on transmission electron microscope because in an electron microscope we assume
or in all microscope we assume that lenses which are being used ok and in an optical
microscope is the one which we are taking it is an a example and talking about all the
lens aberrations i will come to it later and mention to it ah mention to you that the all
the lenses in an electron microscope all the whether its electromagnetic or electro static
lenses these are all convex lenses are the ones which form and no concave lens can be
formed electromagnetic lens because of this the lens aberration especially the spherical
aberration and coma is going to be there similarly ah ah depending upon the design of the ah
lenses how the lenses are machined and formed ok the astigmatism can come so how this astigmatism
is corrected in the case is using what is called as stigmator sigmator is nothing but
in ah plane here because is that the direction in which the ray is passing through we have
some ah plates are rods there to which a positive and the negative voltage is being applied
ok so this will generate essentially a field like this ok these fields what it is going
to be the rays which are away from the optic axis only those rays will be affected and
the rays which are passing close to the optic axis or along the optic axis they are not
going to be disturbed so this is with an electric field we can ah generate a stigmator or we
can keep ah magnets also like this here some magnets are there the north pole in these
two sides and south pole and here again the this is how the field line will be moving
so because of this field lines ok only the peripheral ah rays will be effected by controlling
the strength of this magnetic field we can correct for that astigmatism of the lens ok
and ah what is the ah error which comes due to zone for a point object essentially because
of the astigmatism its ah turns out to be beta is into delta f what is beta is that
if this is the lens size ok and this is the diameter and if we keep an object at the center
what is the maximum angle which submits with the lens this is what this called as the angle
beta the other one as i mentioned the curvature
of the field what is the curvature of that ah field for a lens which will like a planar
complex lens for an object which is ah perpendicular to the line perpendicular to the optic axis
one can see that the rays which are along the optic axis the image is formed at this
particular point obeying the paraxial ray condition or the gaussian formula and the
ray which are far away from it the image is formed at a point which is ah in front of
the gaussian plane so because of that there is a curvature of this field how this can
be corrected this can be corrected by using a lens which is essentially a planar convex
if we use it this will be deviating this rise a little bit so that they will be focused
at the this way we can correct the curvature of the lens
what are the other aberrations of the lens as i mentioned that is because of variation
in magnification across the image field thats a pin cushion barrel distortion spiral distortion
these are all the other aberrations which are lens will have so essentially if we look
at it what we have talked about is for any lens there are five aberrations which are
associated with it what are those aberrations spherical aberration one coma is another astigmatism
is the third curvature of the field is the other aberration distortion is the fifth aberration
ok when we want to compensate for for the aberrations we have to correct each of this
aberrations and make it zero this is what this ah in ah if you go into the theory of
lens aberrations what is called as the sidel terms these terms have to be independently
made equal to zero that means that every aberration has to be that is we cannot use one aberration
to correct for the another so that at a particular point ah it looks like as if the aberration
is corrected in some region each of the aberration has to be independently corrected that is
what it means these are all the aberrations which are connected with the cri with the
lens in addition to it there is an another aberration
which comes ok that aberration which is called as a chromatic aberration ok this also you
might have studied but for the sake of completeness i wanted to mention that this is also one
of the important aberration which has to be corrected that is this aberration arises not
because of any problem with the lens this aberration comes because of non monochromaticity
of the wavelength of the radiation that is the wavelength lambda when we talk about it
this value is not a particular value there is some line with which is associated with
it when there is a width which is associated with the ah radiation ok the as i mentioned
earlier the lens acts its a phase shift that means that depending upon the thickness of
the lens and the wavelength of the radiation ok that is the phase shift which it introduces
is then the refractive index into t the thickness so depending upon different medium the same
ah even if the wavelength ah depending upon different wavelength the phase which it introduces
is going to change ok that is what essentially is the reason for chromatic aberration ok
in the case of a light radiation it is essentially the the spread is arising because of the finite
width of the monochromatic source in the case of an electromagnetic radiation ok the one
the how do you produce monochromatic electrons ok there are many ways which are done one
is ah thermal ionization is used to produce electrons two or cold or hot field emission
that is in thermal ionization what is being done is that filament is taken it is rise
to a very high at temperature so that the electrons are emitted and the electrons are
coming out of the sample they have a finite spread and these electrons are accelerated
to some particular energy ok so this spread inherently in the electron energy initially
that itself gives rise to a spread in the overall total energy and that will be reflected
in the wavelength of the radiation similarly the voltage which we use high voltage which
we are this is also generated from converting an ah d c into an a c in this case also there
are some ripples which are associated with it and that can also give rise to some fluctuations
in the voltage the third point which happens is inelastic scattering that is as the light
radiation passes through the sample it interacts with the material and sometimes some ah energy
of the radiation is lost as it passes through the sample this gives raise to ah electrons
with different energy which is coming out of the sample surface ok so that means that
when the beam which is initially entering is monochromatic but the beam which comes
out ok there is a spread has increased in that energy and this also will give rise to
chromatic aberration so here also what we can make out is that
since the phase shift is going to be [di/different]different for the rays which are ah ah having different
wavelength we will find that depending upon the that energy ok either they can be focused
at this particular point or they can be focused at this point and this finally gives rise
to as in the case of a spherical aberration this gives rise to disc of least confusion
which is coming so the ah diameter of this disc of least confusion is generating related
to beta the angle which it submits with that lens ok multiplied by delta f and this delta
f ah ah depends upon the spherical aberration coefficient c c of the lens into delta e by
e the energy spread ok for a weak lens that c c is f and ah for a ah strong lens the c
c is f by two ok how is this chromatic aberration corrected in the case of an optical microscope
we can use lenses with different refractive index or a combination of them ok they are
called as a chromatic lenses with which for some particular range of wavelength the chromatic
aberrations could be corrected the electrons the only way we can do it is that we can make
the power supplies ok the fluctuation in the voltages are so small ok and ah the ther so
that its a highly stable supply the thermal spread essentially we can ah reduce it by
going for a field emission gun ok and then we can make the sample as thin as possible
so that the ah influence of a ah inelastic scattering on the ah chromatic aberration
could be reduced another is the lenses which we use are electromagnetic lenses where we
apply current by changing the current the focal length of the lens itself changes ok
by making the ah ah lens ah supply the power supply highly stable we can bring about ah
ah reduction in chromatic aberration these are all the ways the chromatic aberration
is corrected so essentially ah any microscope if we look at it ok the aberrations comes
one from the lens which we are using it the other aberration comes from like a chromatic
aberration which comes from the ah source of the radiation itself whether the radiation
is highly monochromatic or not both this facts together decide all the aberrations going
to be there all this can affect the resolution of the instrument this is what we will discuss
it shortly how these variations aberrations have effect the resolution
before we go into it let us look at how this resolution itself is defined ok we know that
ah if we have a screened on which there is a hole is going to be there there is a light
source from which the light is coming this will be forming on the screen the image of
this hole ok if we make the size of this hole small and small and the size become nearer
to that of the wavelength of the radiation which we are using in it then what happens
is that normally it has been observed that ah you dont see a sharp image there are some
ah rings could be seen ok beyond the geometrical image ok this is called as an airy disc ok
so this has been observed long time back itself when people are been observing at it stars
ok so essentially what we can make out is that ah at the center there is a maximum intensity
ok here the same thing which is ah ah shown with respect to a telescope which is being
used this a lens which in front of it at the center maximum intensity and the first minimum
intensity occurs at a particular point ok the separation between this this x ok thats
what essentially is turns out to be this angle alpha which this submits this turns out to
be is equal to lambda by d these derivations one can from the geometry one can work out
ok so essentially what is going to happen is that this x by two equals this is the sort
of a formula which takes place this formula is that for a if we use a spherical aperture
because the lens itself all can be considered as a spherical aperture and in that case that
bessel function has to be used then this factor alpha the angle which submits this turns out
to be one point two two lambda by d ok where d is the ah ah diameter of the aperture so
if this diffraction limited ah for what we have considered earlier is for a one point
source what is the sort of an image which is formed the image itself we can make out
there is a central disc and there is a varying intensity which ah comes so some rings could
be seen outside suppose we have two coherent sources one and an another sources which is
close by ok then depending upon the separation between them it can so happen that ah the
airy disc corresponding to one of them ok and the airy disc corresponding to quiet far
away they can be seen separately but what is the minimum separation at which they could
be seen this a criterion which has been put forward by rayleigh and this criterion essentially
is that if two points sources are ah regarded as just resolved ok when the principle diffraction
maximum of one image coincides with the first minimum of the other ok then using this geometry
one can derive it and then the formula what we get is is that ah the separation between
that objects if is equal to zero point six one lambda by theta ok where theta is the
angle or ah earlier we have used that term beta to represent it by beta that represents
the limit of resolution so this is the best image which we can achieve ok when we use
a lens or when we use an aperture ok and in the case as an optical microscope ah here
we use an another term that denominator n which we call it as the ah numerical aperture
here what i had shown it is that the same when only one object which is considered ok
that is one light source which is being considered which is an incoherent source then this the
way that image appears for a ah aperture whose size is of the same order as the wavelength
of the radiation are closed to it and when the separation between them is very large
this is how both the sources will give rise to individual images and we can resolve them
very clearly at some particular distance ok where the minimum of the one matches with
the maximum of the other they are just resolved and in this case our eye will not be able
to resolve it so this what essentially the images which are being shown corresponding
to these three cases here we can see the images which are resolved here it is just resolved
here its very difficult to make out this is one image other two image ok so this is the
criterion which has been put forward by rayleigh to say that this is the ah resolution of an
equipment now having considered this ah rayleigh criterion
let us now look at what is the effect of spherical aberration on the resolution what does the
spherical aberration do for a point object ok we dont get a point image we get a disc
of least [confu/confusion]confusion ok or in the gaussian image plane this if we this
is going to be the twice ah the diameter of ah instead of a point image we get a disc
with a diameter ok suppose we have ah the same we can continue the two source are there
point sources are separated by a distance which is larger than this if they are there
each one of them will give rise to a disc of least confusion or on the gaussian plane
we will getting an disc we will getting it ok if they are well separated we will be able
to see them as two different objects ok now for some particular distance and this
will be actually a magnified image which we get it and here for some particular separation
ah that is the disc starts overlapping and then we find that only for a some particular
ah separation between these ah objects we are able to see them as separate one that
means that the resolution now compared to rayleigh criterion the lens aberration has
such since it is increasing the for a point object ah it is not giving a point image ok
there is a spread the resolution is becoming poorer and poorer ok the objects which are
separated by larger distance only could be seen very clearly because of the spherical
aberration the same thing happens with the chromatic aberration also because you know
that for a point object we get a disc of least confusion ok at some pattern particular separation
between them the overlap is such that still we are able to resolve them as two individual
ones ok so so for we have considered with respect to lens aberrations both spherical
aberration and chromatic aberration how it is effecting the resolution ok
now if we take a ah light source itself ok if the light source is ah thermionic ah ah
source then whats going to happen the electron emission is taking place from a finite area
of that sample so the current density is one in matters and this also gives rise to some
practical limit and the size of the source which we can use it ok this size of the source
also has an effect on the resolution so this size of the source is given by this formula
which you can go through it ok and understand it essentially the brighter the source that
means that its from a point ah ah region the electrons are emitted with very high intensity
ok then this value of d zero becomes small that is what essentially one should understand
finally when we consider the resolution we have to consider the total resolution thats
what is called as the point to point what is the spread which us going to cross for
two point which are separated by a finite distance it is decided by the spherical aberration
ok chromatic aberration and the then due a inherent ah ah ah property of the source this
is a rayleigh criterion and then another due to a finite size of the source itself in this
we have just taken the square of the ah all the terms are added but this can be done only
when we assume that all this errors are coming due to essentially a gaussian distribution
ok if it is non gaussian for lorentzian then this has to be just a linear addition say
we know that ah ah as far as the lens is concerned this is what essentially the spherical aberrations
which it comes assuming that the beam is ah monochromatic then this term can be assumed
to be zero we assume that the source is a point source then this term also can be taken
to be zero then these are all two terms which are going to effect the resolution ok and
these are all the formulas which has been ah derived ah earlier for the ah rayleigh
criterion for the point source ok if we use an aperture of a or lens of a particular size
what is the spread which is going to cause ah to the image and the another is that spherical
aberration for a point source what is the spread which is going to cause what one should
remember is that here it is one by beta dependence and in this case it is beta to a power of
two dependence ok if we substitute this value and then try to optimize it we will find that
the value of beta is zero point six one this is the optimum lambda by c s into root three
ok the whole to the power of i think one by four ok this is the optimum angle at which
we get the best resolution and what is the best resolution which is possible this d zero
turns out to be zero point nine c s to the power of one by four and lambda to the power
of three by four ok so essentially from this expression we can
make out that a small variation in lambda reduction in lambda can bring about ah large
difference in the resolution that the resolution can become better whereas for c s there is
a large variation in the spherical aberration coefficient is required ok so with this concept
in fact of lot of high voltage microscopes have been constructed ok in these ah microscopes
ok the voltage has been increased up to present day microscopes are available where one can
go upto three million volt ah ah energy so that resolution can be improved considerably
also the ah when the resolution becomes ah high or also the when the energy becomes high
the thickness of the sample which we can useful thickness of the sample which we can examine
also it increases both are an advantage and but the disadvantage essentially is that when
the energy increases lot of radiation damage is produced in that sample so that ah the
microstructure of the sample could be altered during examination of the sample thats one
of the disadvantage microscope ah this one can substitute values and calculate what normally
happens is that here we considered the wavelength which is this order with a c s for a normal
lenses which are of the order of about one to two millimeter ok spherical aberration
coefficient ok then the value of d turns out to be around close to around zero point two
nanometer essentially what it means is that the limit of the resolution of the microscope
is about zero point two five nanometer but actual resolution which we can practically
achieve is hundred times worsen with almost all the lens aberrations which has been optimized
present day microscopes are there where the spherical aberration can be made zero these
aspects i will talk about it at a later class ok so far we have considered the various types
of lens aberrations and all this lens aberrations because of which the resolution which in principle
we should be able to achieve has come down to this particular value in an electron microscope
the magnification is a that is in conventional optical microscope we use optical lenses convex
or con ah concave lenses or a combination of these lenses to achieve ah magnification
in the case of an electromagnetic lens ah in the case of a ah transmission electron
microscopes since electron is the beam ok ah they are affected by either electric field
or a magnetic field they can be reflected by an electric field or an magnetic field
so both these fields could be used ok as lenses ok to magnify but normally represent a microscope
is essentially the magnetic lenses which are used and these are nothing but ah like a solenoid
a coils are bound around this ok then this is the north one is the north and the south
pole ok and this is how the flux lines are going to be there an electron beam which is
parallel to the optic axis passing through the optic axis that will go on diffracted
but the electron beam which is slightly away from the optic axis that will cunder the come
under the action of the magnetic lines and its ah they will feel ah some different forces
ok these forces are ah one is that velocity will be changed in different directions there
is going to be a field lines which are perpendicular to the ah plane of the length and the field
line along the plane of the lens we can take this components and one can find out what
is going to be the trajectory what is essentially which is going to happen is that the one which
is deviated from here it will be moving around like this ok in a helical path and finally
it is being focused to a point that is a lens what it does is the ray which is coming from
here it will be in a helical path it is broad when it comes here what one should understand
that in a convectional optical microscope the image is inverted here it is not an inversion
but there is a rotation which is taking place and the rotation depends upon rotation of
the image depends upon the strength of the magnetic field ok
here that ah how the ah forces on ah electrons ok ah as well as the magnetic flux field lengths
and how the velocities are affected this is being shown and in this particular case one
can immediately make out that if we look at it it looks like a cylindrical geometry like
a solenoide so essentially a cylindrical coordinate system which is being used then if we use
a cylindrical coordinate system then r theta phi are the coordinates which are there so
for that velocity vector as well as the flux vector we can take this coordinates and the
force which is acting on the electron will be given by minus e into v cross b so this
is how this calculations done this is what it is being plotted so the net effect essentially
is that an electron which is deviating from here is brought back to a focus so this solenoide
acts like a lens so this solenoide coil can itself can be surrounded with a soft ion piece
ok so that they capture all the and with that we can have lenses with different types of
shapes this is one pole this gap which can be generated and another is a soft ion piece
which can be attached so that we can have a pole piece with a different type of a gap
ok and to control the current ah one using a circuit ah which gives a stabilized ah power
supply which has to be used and another is that ah since the current are very high it
is ah the lens itself are cooled with water so that ah the magnetic field is highly stable
this is the formula which one can which is ah used to find out the focal length of the
ah lens generally in the case of an electromagnetic
lenses the focal lens are of the order of one or two millimeters when you know in ah
optical lenses the focal lengths are of the order of ah tens of centimeters whereas here
it is one or two millimeters here in this particular table i had just given a comparison
of electrostatic and electromagnetic lenses design because instead of ah electromagnetic
lenses we can use an electrostatic lens also but the advantage with an electrostatic lens
is that there is no image rotation ok and it is light weight and ah power consumption
is less but the ah voltage has to be high highly stable voltage is ah ah necessary and
ah easier is to focus ions but the problem here what happens is that the physical dimension
of the lens turns out to be very high when the beam energy is very high here in these
case the magnetic carved lenses its the smaller dimension which we can use it the aberrations
ah ah can be lowered considerably that is one advantage here the aberrations are high
no high voltage insulation is required whereas for a when ah the power supply becomes very
high ok for high energy electrons then the insulations and all it becomes a massive in
size ok and the another is it can be used as an immersion lens where that ah the sample
itself can be inserted into the lens itself these are all the advantages we will stop
here now thank you

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