# Mod-08 Lec-04 Lens aberrations – Part II

lens aberrations one by m k srivastava department

of physics indian institute of technology roorkee uttharkand in the last lecture we considered chromatic

aberration and spherical aberration in this lecture we shall take up the remaining mono

chromatic aberrations and these are number 1 coma number 2 astigmatism number 3 curvature

of the field and the fourth one distortion let us begin with coma the coma or comatic aberration is a primary

aberration associated with an object point even a short distance away from the axis away

from the axis the main thing its origin lies in the fact that the principal planes can

actually be treated as planes only in the paraxial region they are in fact principle

curved surfaces look at this figure showing grace coming from

first principle focal point the emergent ray is naturally our parallel but the point of

intersection of the race diverging from the first principle focal point and the emergent

parallel rays and their point of intersection does not lie on a plane it is really curved

particularly mainly actually for those days which are not paraxial similarly in the second

figure there is l is after refraction they are focused to the second focal point but the point of intersection of the incident

parallel rays and the refracted rays which are getting focused at the circle focal point

their intersection does not lie on a plane surface that surface is curved this curvature

appears really for those rays which are not paraxial for the marginal rays the rays we proceed near the axis of the lens

they focus at a point different from that of the marginal rays thus it appears that

the magnification is different for different parts of the lens if you consider the image

formation by different zones of a lens the paraxial zone or the marginal zone then the

spherical aberration arises due to the fact the different zones have different powers

and coma it arises due to the fact the different zones have different magnifications the figure illustrates the effect of a coma

the resultant image of distant point object of the axis is shown in the side figure the

rays in the tangent plane are shown you see the chief ray which passes through the center

of lens goes undeviating and you see the marginal rays these are the ones which are which give

rise to coma a broader circle near the axis and the vertex close to the point p let us consider in this figure several skew

rays are shown from an extra axial point note that the each circular cone of rays whose

end points 1 2 3 4 that a circular zone of the lens from where the rays are coming they

are focused in a in a chromatic circle you see at the radius of the circular zone decreases

as we come towards the center of the lens this cone on the image the comatic circle

this radius goes also goes on becoming smaller and smaller and it centered moves towards

the vertex of the cone in the limit when we consider the rays passing through the center

of the lens that gives rise to the vertex of the cone now this case corresponds to a positive gamma

so that larger the ring on the lens the more distant its cometic circle from the axis now gamma can be eliminated if a lens satisfies

the abbe’s sine condition n1 y1 sine theta1=n2 y2 sine theta 2 n1 y1 theta1 they refer

to the refractive index height of the object above the axis and the slope angle of the

incident ray of light similarly n2 y2 and theta 2 they refer to the corresponding quantities

in the image medium they made site now the magnification of the image is given by y2

divided by y1 which according to this condition should be equal to n1 sine theta1 divided

by n2 sine theta2 now the elimination of coma is possible if

the lateral magnification y2 upon y1 is same for all the rays irrespective of the slope

angle the respective of theta 1 and theta 2 thus coma can be eliminated if sine theta1

divided by sine theta 2 is a constant a lens that satisfies this condition is called an

aplanatic lens such a lens is mostly used at the front lens of a high powered microscope

objective called a oil immersion objective it is the oil chosen is naturally such that

it has the same refractive index as that of the lens the oil is placed on a microscope

slide which is being observed and then the lenses lowered into contact contact with the

oil fill in this figure shows the aplanatic lens is

all raised from an object that m leave the hemispherical surface after refraction as

they though they came from the point m prime which is the first image on the spherical

surface and this introduces a lateral magnification and primary divided by ma now if the second

lens is added which has the center of its curvature called of its concave surface at

m prime and therefore is the surface is normal to

all rays the reflection at its upper surface of ready of radius m prime times cm prime

will give an additional magnification without introducing a spherical aberration and the

final image is formed at m double prime now this property of the upper lens however

holds strictly only for rays from a single point m and not for points adjacent to it

that is a limitation of this arrangement now like it is very collaboration coma is

dependent on the shape of the lens but the interesting thing is the fact that it can

be made exactly zero for the single lens with a given object distance this is important

with a given object the particular shape it then has is almost convex planar one side

convex other side almost plane and this shape if you remember it also nearly the shape for

minimum spherical aberration as we have seen in the last lecture is this

figure shows the variation of the spherical aberration and also the variation of coma

for different shape factors if you remember for a shape factor about +5 the spherical

aberration is minimum and that is also the region where the coma passes through the 0

axis so that is a very interesting thing at the same time you can eliminate this coma

and minimize spherical aberration by proper choice of the radii r1 and r2 now we come to astigmatism now if the first

two seidel sums vanish all rays from points on are very close to the axis of a lens will

form point images and there will be no spherical aberration or coma now when an object point lies an appreciable

distance from the optical axis the incident cone of rays will strike the lens symmetrically

giving rise to a third primary aberration known as astigmatism the third seidel sum

s3 is not therein the case of an axial point the cone rays is symmetrical with respect

to the spherical surface of the lens there is no need to make a distinction between maridional

and sagittal planes the ray configurations in all planes containing

the optical axis or identical there in contrast here the configuration of an oblique parallel

ray will be different in the maridional and sagittal planes as a result before collins

in this principle in these planes will be different in effect here the maridional rays

are tilted more with respect to the lens than or the sagittal planes rays and they have

therefore the shorter focal length this astigmatic difference increases rapidly

as the rays become more and more oblique that is as the object point moves further of the

axis and is naturally 0 when the object is on the axis having two different focal lens the incident

conical bundle of rays diverging from the object takes on a considerably altered form

after reflection the rays in the maridional plane converged at a different point as compared

to those in this astigmatic plane now this figure shows raise diverging from

the point p in the maridional plane constituted by rays pa and pb they are focused at the

point t while the rays for the sagittal plane constituted by rays pc and pd are focused

at the point p their focus little farther away at the point s the rays pa and pb in

the figure focus at the point t and rays pc and pd focus at a point s which is different

from t as shown in the figure now since at the point t the rays in the sagittal

plane have not still focused one in fact has a focal line which is normal to the maridional

plane this vocal line t is called the initial focus maridional focus are the primary image now beyond this point the beams cross section

rapidly opens out until it is again almost a circle at this location image in a circular

blur not very well focused naturally and known as the circle of least confusion moving further at s rays in the maridional

plane have now defocused one again obtains a focal line lying in the tangential plane

and this is called the sagittal focal line or the secondary image it is in the maridional

plane the distance between s and t is a measure of astigmatic this figure shows that variation at p1 this

is what in the figure was the point t the tangential image where the rays in the maridional

plane are focused then as we move away from it away from the lens the beam bundle of beam

opens up opens up a little more becomes circular then again begins to close in in a perpendicular

direction close it further and at the point p2 you get the image where the sagittal rays

are in focus one thing should be noted the secondary line

image will change in orientation the changes in the object position but it will always

point towards the optical axis that is it will be radial that is the interesting thing

similarly the primary image where the maridional rays came in focus this line image will vary

in orientation but it will remain normal to the secondary image you see this features causes a very interesting

effect when object is made up of radial radial and tangential elements the primary and secondary

majors are in fact formed of the transverse and radial dashes let us look at this little

carefully this is where we are interested in forming the image of an object which has

got a rim and radial spokes okay after passing through the lens if we consider here in the

tangential focal plane t this is where the maridional rays have come in focus this is the transverse image so the rim is

getting focused the spokes they are very blurred but then as we move it to the sagittal focal

plane now in this case we are getting the radial spokes are focused and the tangitial

rim has gone out of focus now to analyze the origin of astigmatic 1

points that for a point on the axis and the lens is free from other aberrations the wave

fronts emerging from the lens are the spherical and us as the wave front progresses that is

moves it converges to a single point but when the object point is non-axial quite away from

the axis then the emerging wave front is not a spherical anymore and as the wave front

converges it does not focus to a point but to two lines which are normal each other

the maridional line and sagittal line now somewhere between these two lines image is

circular and is called the circle of least confusion but is it is still not too very

well focus there now the distance between the tangential foci

and sagittal foci increase as the object point moves away from the axis thus the tangential

foci and sagittal foci of the points at different distances from the axis lie on two surfaces

the optical system is said to be free from astigmatism when the two surfaces coincide

they will still be curved but we shall come to that a little later one thing we know these

two surfaces they are coinciding at the optical axis of the system but naturally that the axial point this is

the problem of astigmatism is there as we move away from the axis and as i pointed out earlier even if we surfaces

coincide okay the sd committal will not be there but the resultan image surface is in

general curve and this defect of image is called curvature of the field now the shape of the image surface depends

on the shape of the lens and the position of the stops if the primary surface is to

the left of the secondary image astigmatism is said to be positive otherwise negative

now by using a convex and concave lens of suitable focal lengths and separated by a

distance it is possible to minimize the astigmatic difference such a lens combination is called

astigmatism now we come to another defect and the reparation called curvature the image of an extended plane object due

to a single lens is not a flat one but is a curved surface the central portion of the

image nearer the axis is in focus but the outer region of the image away from the axis

those regions are blurred this defect is called the curvature of the field it is due to the

defect that the paraxial focal length is greater than the marginal focal length marginal focal length means the focal length

for the marginal rays this separation is present even if the aperture of the lens is reduced

by a suitable stop usually employed to reduce a spherical aberration coma and astigmatism you see a real image formed by a convex lens

curves toward the lens if the points a and b are mirrored curves towards the lens for the case of a virtual image which is formed

at a distance larger than that of the object a prime and b prime these are curved away

from the lens this figure shows the curvature of the field

in the case of a concave lens here again the images formed nearer the ends and the ends

a prime and b prime or curved toward lens now for the system of thin lenses to get a

flat image this is the condition summation over 1 upon ni fi summed over the various

lens components for two lenses this condition reduces to 1 upon n1 f1 + 1 upon n2 f2=0

this is known as petzwal’s condition for no curvature this condition holds good whether the lenses

are separated by a distance or placed in contact and the refractive indices are positive the

above condition will be satisfied if you lenses are of opposite sign if one lens is convex

the other must be concave now individual instruments a certain amount

of curvature can be tolerated is not very serious the reason is the eye can accommodate

for it however in photographic lenses field curvature most undesirable since it has the

effect of rapidly blurring the off-axis image the curvature of the fluid can be corrected

for single lens by means of a stop now we come to the last one of mono chromatic

aberrations and that is distortion the variation in the magnification produced by a lens for

different actual distances results in an aberration called distortion to be free of distortion

the system must have uniform little magnification over its entire field a pinhole camera is

ideal in this respect what it shows no distortion at all all the state lines connecting each

pair of conjugate points in the object image planes they pass through the opening the constant magnification for a pinhole camera

as well as for a sim lens implies then phi prime upon tan phi should be a constant if

this figure shows several ways mm prime is the actual ray then for the ray q1 upon q2

q1 q1 prime the angle is phi1phi1 prime this ratio is same as for the ray q2 q2 prime and

the angles are phi2 and phi2 prime common forms of image distortion in an actual

lens system are shown in the figure here first one represents the undistorted image of an

object consisting of a regular wire mesh the next diagram shows the barrel distortion this

arises when the magnification decreases towards the edge of the field it is more near the

center of the field and the third figure represents a pincushion distortion corresponding to a

greater magnification at the borders leading to looks like a stretched configuration a single thin lens is particularly free of

distortion for all object distances it cannot however be free for all the other aberrations

at the same time now if a stop is placed before the lens that is in the object space the distortion

is barrel shaped more magnification near the axis and less towards the edges and if i stop

is placed after the lens that is in the image region the distortion is pincushion type stretching

at the edges so that immediately gives us an idea to eliminate

distortion is top is usually placed in between two symmetrical lenses so that the pincushion

distortion produced by the first lens compensated by the barrel shape distortion produced by

the second lens projection and camera lenses are constructed in this way now let us summarize these things briefly

all the aberrations we have gone through the various methods of collecting for on a chromatic

aberrations spherical aberration and coma we have seen can be corrected by using a contact

doublet of the proper shape astigmatism and curvature of the field require for the correction

the use of several separated components the distortion can be minimized by the proper

placement of a stop in between this mess the components of a lens system now let us see this problem in totality and

keeping in mind different types of optical instruments the lens may be affected by as

many as 7 primary abrasions 5 mono chromatic aberrations as we have seen and 2 chromatic

aberrations longitudinal and lateral now one might therefore wonder how it is possible

to make a good lens at all when rarely can single aberration be eliminated completely

much less all of them at the same time simultaneously this is the good usable insular nevertheless

made by the proper balancing of the various aberrations you see the design is guided by

the purpose for which the lens is to paint you see that is the main thing where the lenses

to be used then guides which aberrations are to be eliminated or minimize or properly to

be taken care of in a telescope objective for example correction

for chromatic aberration spherical aberration and coma are of primary importance on the

other hand astigmatism curvature of the field and distortion are not a serious because the

field over which the objective is to be used is relatively small for a good camera lens

of wide of aperture wide angle the field situation is almost exactly reverse in case i think

this we have come to the end of this lecture series hope you have liked it thank you