UNIT-1�CHAPTER-2 : Lens Aberrations

1. INTRODUCTION

One of the basic problem of lenses is the imperfect quality of the images. The simple equations derived and discussed in the earlier chapters, connecting object and image distances, focal length etc are based on the assumption that the angles made by the rays of light with the axis are small and paraxial approximation may be made. In actual practice the objects are bigger and a lens is required to produce a bright and magnified image. We are therefore required to take into consideration the wide-angle rays from the centre of the object and also the upper and lower parts of the object and falling near the top and bottom of the lens. These rays are known as peripheral or marginal rays.

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In general, peripheral rays of light do not meet at a single point after refraction through the lens. Secondly, the refractive index and hence the focal length of a lens is different for different wavelengths of light. For a given lens, the refractive index for violet light is more than that for red light. Thus, if the light coming from an object point is not monochromatic, the lens forms a number of coloured images. These images, even though formed by paraxial rays, are at different positions and are of different sizes.

2. ABERRATIONS

The departures of real images from the ideal images, the respect of the actual size, shape and position are called aberrations. In other words, an aberration is any failure of a mirror or lens to behave precisely according to the simple formulae we have derived. Aberrations are only due to inherent shortcomings of a lens and not caused by faulty construction of the lens, such as irregularities in its surfaces. They are inevitable consequences of the laws of refraction of spherical surfaces.

Aberrations are divided broadly into two categories- monochromatic aberrations and chromatic aberrations. The defects due to wide-angle incidence and peripheral incidence, which occur even with monochromatic light are called monochromatic aberrations. Aberrations that occur due to dispersion of light are called chromatic aberrations. Chromatic aberration occurs with light that contains at least two wavelengths

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Monochromatic aberrations are again divided into five types:

1. Spherical aberration

2. Coma

3. Astigmatism

4. Curvature of field

5. Distortion.

The deviations from the actual size, shape and position of an image as calculated from the earlier simple equations are called the aberrations produced by a lens. The aberrations produced by the variation of refractive index with wavelength of light are called chromatic aberrations. The other aberrations are caused even if monochromatic light is used and they are called monochromatic aberrations. Lens aberrations are just the consequence of the refraction laws at the spherical surfaces and not due to defective construction of the lens such as the surfaces being not spherical etc.

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3. FIRST ORDER THEORY

To understand satisfactorily the theory of lens aberrations, it is necessary to start with the expansion of the series of angles into a power series. According to Maclaurin’s theorem the expansion of sin θ is given by

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When the value of θ is small, the series is a rapidly converging one i.e., the value of any term is smaller than the preceding one. In case the slope angle is small, sin θ = θ approximately. The equations developed on the basis that the sines of the angles are equal to the angles from the basis of the first order theory.

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In Fig. 1 for small values of θ the height of the perpendicular AC can be taken approximately equal to the length of the arc AB.

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Fig. 1

Table 1 gives the variation of sin θ with increasing angle.

The difference in the values of sin θ and is much smaller than sin θ and θ.

TABLE 1

4. THIRD ORDER THEORY

If in the formulae for reflection and refraction at spherical surfaces the first two terms of the series are replaced for values of sines of angles, the results obtained represent the third order theory. The formulae thus obtained give a fairly accurate account of the principal aberrations. In the third order theory, the aberration of a ray of light, viz., its deviations from the path obtained from Gauss formulae, is denoted by five sums called the seidel sums. A lens will be free from all the aberrations, if all the five sums are equal to zero. But In practice, no optical system can be made to satisfy all the conditions at the same time. Let S₁, S₂ etc denote the five seidel sums. Then spherical aberration is absent if S₁ = 0 ; coma is absent if S₁ = 0 and S₂ = 0; astigmatism and curvature of the field are absent if S₁ = 0, S₂ = 0, S₃ = 0 and S₄ = 0. Finally if S₅ is also equal to zero the image of an axial object will be free from distortion as well. These five defects of an image are called the monochromatic aberrations.

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5. SPHERICAL ABERRATION

A lens may be regarded as made

up of a large number of Prisms, of

increasing angles from the centre to

outward in case of a convex lens and of decreasing angles in case of concave lens. A ray of light falling on a prism of larger angle is deviated more towards the base of the prism than that falling on a prism of small angle. Therefore, peripheral light rays passing through a lens farther away from the axis are refracted more and come to focus closer to the lens. Paraxial rays passing through the lens close to the axis are refracted less and come to focus farther from the lens. Therefore, rays passing through different zones of a lens surface come to different foci. An image formed by paraxial rays will be surrounded by a diffuse halo formed by peripheral rays and consequently the image is blurred. This phenomenon is known as spherical aberration.

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Spherical Aberration : Causes halos around points of light

The presence of spherical aberration in the image formed by a single lens is illustrated in Fig.1. O is a point object on the axis of the lens I_p

Fig. 1

and I_m are the images formed by the paraxial and marginal rays respectively. It is clear from the figure that the paraxial rays of light from the image at a longer distance from the lens than the marginal rays. The image is not sharp at any point on the axis. However, if the screen is placed perpendicular to the axis at AB, the image appears to be a circular patch of diameter AB. At positions on the two sides of AB, the image patch has a larger diameter. This patch of diameter AB is called The circle of least confusion, which corresponds, to the position of the best image. The distance I_m I_p measures the longitudinal spherical aberration. The radius of the circle of least confusion measures the lateral spherical aberration. When the aperture of the lens is relatively large compared to the focal length of the lens, the cones of the ray of light refracted through the

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different zones of the lens surface are not brought to focus at the same point I_m and the axial rays comes to focus at a farther point I_p. Thus, for an object point O on the axis, the image extends over the length I_m I_p. This effect is called spherical aberration and arises due to the fact that different annular zones have different focal lengths. The spherical aberration produced by a concave lens is illustrated in Fig. 2

Fig. 2

The spherical aberration produced by a lens depends on the distance of the object point and varies approximately as the square of the distance of the object ray above the axis of the lens. The spherical aberration produced by a convex lens is positive and that produced by a concave lens is negative.

5.1 REDUCING SPHERICAL ABERRATION

Spherical aberration produced by lenses is minimized or eliminated by the following methods.

1. Spherical aberration cab be minimized by using stops , which reduce the effective lens aperture. The stop used can be such as to permit either the axial rays of light or the marginal rays of light. However, as the amount of light passing through the lens is reduced, correspondingly the image appears less bright.
2. The longitudinal spherical aberration produced by a thin lens for a parallel incident beam is given by

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where x is the longitudinal spherical aberration, ρ is the radius of the lens aperture and f₂ is the second principal focal length.

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Where R₁ and R₂ are the radii of curvature. For given values of μ, f₂ and ρ, the condition for minimum spherical aberration is

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Differentiating equation (1) and equating the result to zero, we get

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From equation (2), for a lens whose material has a refractive index μ = 1.5, Thus, the lens, which produces minimum spherical aberration, is biconcave and the radius of curvature of the surface facing the incident light is one-sixth the radius of curvature of the other face.

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Fig. 1

In general, the more curved surface of the lens should face the incident or emergent beam of light which ever is more parallel to the axis. A lens whose is called a cross lens. The process in which the shape of the lens is changed without changing the focal length of the lens is called bending of the lens for minimum spherical aberration. A crossed lens is shown in Fig. 1.

It is clear from the figure that the deviation produced by the two surfaces is the same and the axial and marginal rays of light come to focus with minimum of spherical aberration. However, it should be noted that the spherical aberration cannot be completely eliminated in a lens with spherical surfaces.

For a lens of refractive index 1.5, focal length 100 cm and radius of the lens aperture 10 cm, the longitudinal spherical aberration is 1.07 cm for k = - 1/6. For the same values of focal length and radius of the lens aperture, if the values of μ and k are 2 and + 1/5, the longitudinal spherical aberration reduces to 0.44 cm.

Plano-convex lens

3. Plano-convex lenses are used to optical instruments so as to reduce the spherical aberration. When the curved surface of the lens faces the incident or emergent light whichever is

more parallel to the axis, the spherical aberration is

minimum. The spherical aberration in a crossed lens

is only 8 % less than that of a Plano-convex lens having

the same focal length and radius of the lens aperture. This is the reason why plano-convex lenses are generally used in place of crossed lenses without increasing spherical aberration appreciably. Fig. 2 represents the variation of longitudinal spherical aberration with the radius of the lens aperture for lenses of the same focal length and refractive index.

The spherical aberration will, however be very large if the plane surface faces the incident light. The spherical aberration is a result of larger deviation of the marginal rays than the paraxial rays. If the deviation of the marginal rays of light is made minimum, the focus f_m for a parallel incident beam will shift towards f_p, the focus for the paraxial rays of the light and the spherical aberration will be minimum.

Fig. 2

Fig. 3

Fig. 4

As the deviation is a minimum in a prism, when the angles of incidence and emergence are equal, similarly in a lens also, spherical aberration can be minimized if the total deviation produced by a lens is equally shared by the two surfaces. In a plano-convex lens, when the plane surface faces the parallel beam of light, the deviation is produced only at the curved surface and hence the longitudinal spherical aberration (Fig. 3) is more than when the curved surface faces the incident light (Fig. 4). In the latter case the spherical aberration is less than the former, because the total deviation in the second case is divided between the two surfaces. Thus, the spherical aberration produced by a single lens cab be minimized by choosing proper radii of curvature. The shape factor q of lens is given by

Fig. 5

Fig . 5 represents lenses of the same values of f, μ and ρ but the different shape factors calculated according to equation (3). The longitudinal spherical aberration is represented along the y-axis and the shape factor q along the x-axis.

Spherical aberration for a double convex lens (shape factor 0.5) is a minimum when the surface of smaller radius of curvature faces the incident parallel light. The spherical aberration for a plano-convex lens (shape factor +1.0) when the curved surface faces the incident light is only slightly more than the double convex lens. Hence, plano-convex lenses are preffered.

4. Spherical aberration can also be made minimum by using two plano-convex lenses separated by a distance equal to the difference in their focal length. In this arrangement, the two lenses equally share the total deviation and the spherical aberration is minimum. In Fig. 6, two plano-convex lenses of focal lengths f₁ and f₂ are separated by a distance d.

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Fig. 5

Let δ be the angle of deviation produced by each lens (see Fig. 5).

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And from Δ^lesBGF₁, BG=GF₁ or O₂G = GF₁ (approximately)

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For the second lens, F₁ is the virtual object and G is the real image. Substituting these values of object and image distances in the formula.

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Thus, the condition for minimum spherical aberration is that the distance between the two lenses is equal to the difference in their focal lengths.

5. Spherical aberration for a convex lens is +ve and that for a concave lens is –ve. By a suitable combination of convex and concave lenses spherical aberration can be made minimum.

6. Spherical aberration may be minimized by using axial-GRIN lenses.

6. COMA

The effect of rays from an object point not situated on the axis of the lens results in an aberration called coma. Comatic aberration is similar to spherical aberration is that both are due to the failure of the lens to bring all rays from a point object to focus at the same point. Spherical aberration refers to object points situated on the axis whereas comatic aberration refers to object points situated off the axis. In the case of spherical aberration, the image is a circle of varying diameter along the axis and in the case of comatic aberration the image is comet-shaped and hence the name coma.

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Fig. 1

Fig. 1 illustrates the effect of coma. The resultant image of a distant point off the axis is shown on the right side of the figure. The rays of light in the tangential plane are represented in the figure.

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Fig. 2 illustrates the presence

of coma in the image due to a

point object, O, situated of the axis of the lens. Rays of light getting refracted through the centre of the lens (ray 1) meet the screen XY at the point P. Rays 2,2; 2,3 etc getting refracted through the outer zones of the lens come to focus at points Q, R, S, etc, nearer the lens and overlapping circular patches of gradually increasing diameter are formed on the screen. The resultant image of the point is comet-shaped as indicated on the right side of the figure.

Fig. 2

Fig. 3

Let 1,2,3 etc be the various zones of the lens (Fig. 3a). Rays of light getting refracted through these different zones give rise to circular patches of light 1’, 2’, 3’, etc. The screen is placed perpendicular to the axis of the lens and at the position where the central rays come to focus [Fig. 3b]. Like spherical aberration, cometic aberration produced by a single lens can also be corrected by properly choosing the radii of curvature of the lens surface. Coma can be altogether eliminated for a given pair of object and image points whereas spherical aberration cannot be completely

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corrected. Further, a lens corrected for coma will not be free from spherical aberration and the one corrected for spherical aberration will not be free from coma. Use of a stop or a diaphragm at the proper position eliminates coma.

coma is the result of varying magnification for rays refracted through different zones of the lens. For example, in Fig. 2, rays of light getting refracted through the outer zones come to focus at points nearer the lens. Hence the magnification of the image due to outer zones is large than the inner zones and in this case coma is said to be positive. On the other hand if the magnification produced in an image due to the outer zones is smaller, coma is said to be negative.

According to Abbe, a German optician, coma can be eliminated if a lens satisfies the Abbe’s sine condition viz.

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Where μ₁, y₁ and θ₁ refer to the refractive index, height of the object above the axis and the slope angle of the incident ray of light respectively. Similarly μ₂, y₂ and θ₂ refer to the corresponding quantities in the image space. The magnification of the image is given by .

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Elimination of coma is possible if the lateral magnification y₂/y₁ is the same for all rays of light, irrespective of the slope angles θ₁ and θ₂. Thus, coma can be eliminated if is a constant because is constant. A lens that satisfies the above condition is called an aplanatic lens.

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6.1 APLANTIC POINTS

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Aplanatic lens.

A spherical lens, which is free from the defects of spherical aberration and coma is called an aplanatic lens. A pair of conjugate points free from spherical aberration and coma is called aplanatic points. Fig. 4 illustrates the property of an aplanatic lens.

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Fig. 4

Let O be the center of curvature of the lens of refractive index μ and radius of curvature R. P is a point on the axis of the lens such that

PO = R/μ. It can be shown that all rays passing through the point P appear to diverge through the point Q irrespective of the slope angle made by the incident rays. PA is the incident ray and AC is the refracted ray. The ray AC appears to diverge from the point Q which is the image of P. Let i and r be the angles of incidence and refraction and α and β the slope angles made by the incident and refracted rays.

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Then,

• OQ = μ R

Thus, if the distance of the object point P is R / μ from the centre of curvature, then distance of the image point Q is μ R irrespective of the slope angles α and β. The object and image distances or the conjugate points that satisfy the Abbe condition (1) can be seen from the Fig. 4 as

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The object and image points which are at distances given by equ. (6) from the pole of the spherical surface are free from spherical aberration as well as coma. Such points are known as aplanatic points.

As aplanatic lens is mostly used as the front lens of a high power microscope objective called the oil immersion objective. As it is not possible to place an object inside a solid spherical lens, the lens is ground little and the object to be examined is embedded in between a drop of oil and the lens surface. The oil chosen in such that it has the same refractive index as that of the lens.

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7. ASTIGMATISM

Astigmatism, similar to coma, is the aberration in the image formed by a lens, of object point off the axis. The difference between astigmatism and coma, however, is that in coma the spreading of the image takes place in a plane perpendicular to the lens axis and in astigmatism the spreading takes place along the lens axis. Astigmatism discussed in this article the different from the one treated in defective version.

Fig. 1

Astigmatism

Fig. 1 illustrates the defect of astigmatism in the image of a point B situated off the axis. Two portions of the cone of rays of light diverging from the point B are taken. The cone of the rays of light refracted through the tangential (vertical) plane BMN comes to focus at point P₁ nearer the lens and the cone of rays refracted through the sagittal (horizontal) plane BRS come to focus at the point P₂ away from the lens. All rays pass through a horizontal line passing through P₁ called the primary image and also through a vertical line passing through P₂ called the secondary image.

The refracted beam has an elliptical cross-section, which ends to a horizontal line at P₁ and a vertical line at P₂. The cross section of the refracted beam is circular at some point between the primary and the secondary images and this is called the circle of least confusion. If a screen is held perpendicular to the refracted beam between the points P₁ and P₂, the shape of the image at different positions is as shown in Fig. 2.

Fig. 2

The locus of the primary images of all points in the object plane gives the surface of revolution about the lens axis and is called the primary image surface. The locus of the secondary images gives the secondary image surface. The surface of best focus is given by the locus of the circles or least confusion. The primary and the secondary image surfaces and the surface of best focus are illustrated in Fig. 3. P₁ and P₂ are the images of the object point B.

Fig. 3

TPN and SPR are the first and the second image surfaces and KPL is the surface of best focus. The three surfaces touch at

the point P on the axis. Generally, the surface of best focus is not plane but curved as shown.

This defect is called the curvature of the field. The shape of the image surfaces depends on the shape of the lens and the position of the stops. If the primary image surface is to the left of the secondary image surface, astigmatism is said to be positive, otherwise negative. By using a convex lens of suitable focal lengths and separated by a distance, it is possible to minimize the astigmatic difference and such a lens combination is called anastigmat .

8. CURVATURE OF THE FIELD

The image of an extended object due to a single lens is not a flat one but it will be a curved surface. The central portion of the image nearer the axis is In focus but the outer regions of the image away from the axis are blurred. This defect is called the curvature of the field.

This defect is due to the fact that the paraxial focal length is greater than the marginal focal length. This aberration is present even if the aperture of the lens is reduced by a suitable stop, usually employed to reduce spherical aberration, coma and astigmatism.

Fig. 1

Fig. 1 illustrates, the presence of curvature of the field in the image formed by a convex lens. A real image formed by a convex lens curves towards the lens [Fig. 1 (a)] and a virtual image curves away from the lens [Fig. 1 (b)]. Fig. 2 represents the curvature of the field present in the image formed by a curvature of the field present in the image formed by a concave lens.

Fig. 1

Fig. 2

For a system of thin lenses, the curvature of the final image can, theoretically, be given by the expression

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Where R is the radius of curvature of the final image, μ_n and f_n are the refractive index and focal length of the n^th lens. For the image to be flat, R must be infinity.

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Correspondingly, the condition for two lenses placed in air, reduces to

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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.

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As the refractive indices μ₁ and μ₂ are positive, the above condition will be satisfied if the lenses are of opposite sign. If one of the lenses is convex, the other must be concave.

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Astigmatism and coma are completely eliminated if the primary and secondary image surfaces are coincident and plane. In this case, the surface of best focus will also be a plane one. But this cannot be achieved with a single lens.

Fig. 3

Fig. 4

Astigmatism or curvature of the field can be minimized by introducing suitable stops on the lens axis. If the primary and the secondary image surfaces are made to have equal and opposite curvatures (Fig. 3), the surface of best focus will be plane and midway between them. Astigmatism will however, be present. Astigmatism can be eliminated by having the same curvature for the primary and secondary image surfaces.

In this case curvature of the field will be present (Fig. 4). Correction for coma is more important than astigmatism for object point having comparatively small angular distances from the axis. Hence, telescope objectives, whose field of view is small, are corrected for coma rather than for astigmatism. On the other hand a camera lens of wide field has to be necessarily corrected for astigmatism.

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9. DISTORTION

The failure of a lens to form a point image due to a point object is due to the presence of spherical aberration, coma and astigmatism. The variation in the magnification produced by a lens for different axial distances results in an aberration called distortion. This aberration is not due to the lack of sharpness in the ‘image’. Distortion is of two types viz., (a) pin-cushion distortion and (b) barrel-shaped distortion.

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Fig. 1

In pin-cushion distortion, the magnification increases with increasing axial distance and the image of an object [Fig. 1 (a)] appear as shown In Fig. 1(b). On the other hand, if the magnification decreases with increasing axial distance, it results in barrel-shaped distortion and the image appears as shown in Fig.1(c).

In the case of optical instruments intended mainly for visual observation, a little amount of distortion may be present but it must be completely eliminated in a photographic camera lens, where the magnification of the various regions of the object must be the same.

In the absence of stops, which limit the cone of rays or light striking the lens, a single lens is free from distortion. But, if stops are used, the resulting image is distorted. If a stop is placed before the lens the distortion is barrel-shaped [Fig. 2(a)] and if a stop is placed after the lens,

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the distortion is pin-cushion type [Fig.2(b)]. To eliminate distortion a stop is placed in between two symmetrical lenses, so that the pin-cushion distribution produced by the first lens is compensated by the barrel-shaped distortion produced by the second lens [Fig. 2(c)]. Projection and camera-lenses are constructed in this way.

Fig. 2

10. CHROMATIC ABERRATION

The refractive index of the material of a lens is different for different wavelengths of light. Hence the focal length of a lens is different for different wavelengths. Further, as the magnification of the image is dependent on the focal length of a lens, the size of the image is different for different wavelengths (colours). The variation of the image distance from the lens with refractive index measures axial or longitudinal chromatic aberration and the variation in the size of the image measures lateral chromatic aberration.

Fig. 1

Fig. 1 illustrates chromatic aberration present in an image formed by a single lens L. AB is an object placed in front of the lens. A’B’ and A”B” are the violet and the red images. The violet image is formed nearer the lens than the red image. The monochromatic aberration are assumed to be absent in this case.

The distance x measures the axial or longitudinal chromatic aberration and the distance y measures the lateral chromatic aberration. The images of intermediate colours between violet and red lie in between the images A’B’ and A”B” and their size increases from violet to red. At no one position the images are in sharp focus. Thus, a single lens produces a coloured image of an object illuminated by white light and the defect is called chromatic aberration. Elimination of this defect in a system of lenses is called achromatism.

11. CHROMATIC ABERRATION IN A LENS

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