# 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