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chapter four
THIN LENSES
Types of lenses
A thin lens may be defined as one whose thickness is considered small in comparison with the distances generally associated with its optical properties. Such distances are, for example, radii of curvature of the two spherical surfaces, primary and secondary focal lengths, and object and image distances.
The combination of various surfaces of thin lenses will determine the signs of the corresponding spherical radii.
4.1 FOCAL POINTS AND FOCAL LENGTHS
Diagrams showing the refraction of light by an equiconvex lens and by an equiconcave lens are given in figure 4.2. The axis in each case is a straight line through the geometrical center of the lens and perpendicular to the two faces at the points of intersection. For spherical lenses this line which joins the centers of curvature of the two surfaces. Ray diagrams shown in the figure 4.2 illustrates the primary and secondary focal points F and F' and the corresponding focal lengths f and f' of thin lenses.
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4.2 CONJUGATE POINTS AND PLANES
If the principle of the reversibility of light rays is applied to figure 4.3, we observe that Q'M' becomes the object and QM becomes its image. The object and image are therefore conjugate. Any pair of object and image points such as M and M' in figure 4.3 are called conjugate points , and planes through these points perpendicular to the axis are called conjugate planes.
Figure 4.3 Image formation by an ideal thin lens. All rays from an object point Q which pass through the lens are refracted to pass through the image point Q'.
If we know the focal length of a thin lens and the position of an object, there are three methods of determining the position of the image : (1) graphical construction, (2) experiment, and (3) use of the lens formula
Here s is the object distance, s' is the image distance, and f is the focal length, all measured to or from the center of the lens. This lens equation will be derived later in this chapter.
4.3 Sign Conventions for Thin Lenses 1: object distance ( s ) is positive if object is in front of lens and is a negative if object is in back of lens. 2: image distance ( s' ) is positive if image is in back of lens and is negative if image is in front of lens.
3: r 1 and r 2 are positive if center of curvature is in back of lens. 4: r 1 and r 2 are negative if center of curvature is in front of lens.
parallel to the axis and will cross the others at Q' as shown in the figure. The numbers I, 2, 3, etc., in figure 4.2 indicate the order in which the lines are customarily drawn.
Figure 4.4a: The parallel-ray method for graphically locating the image formed by a thin lens.
b: For Concave Lens With the negative lens shown in figure 4.4b the image is virtual for all positions of the object, is always smaller than the object, and lies closer to the lens than the object. As is seen from the diagram, rays diverging from the object point Q are made more divergent by the lens. To the observer's eye at E these rays appear to be coming from the point Q' on the far side of but close to the lens. In applying the lens formula to a diverging lens it must be remembered that the focal length j is negative.
Figure 4.4b: The parallel-ray method for graphically locating the image formed by a concave lens.
4.5 LATERAL MAGNIFICATION
A simple formula for the image magnification produced by a single lens can be derived from the geometry of figure 4.4a. By construction it is seen that the right triangles QMA and Q'M'A are similar. Corresponding sides are therefore proportional to each other, so that
where AM' is the image distance s' and AM is the object distance s. Taking upward directions as positive, y = MQ, and y' = -M'Q'; so we have by direct substitution y'/y = -s'/s. The lateral magnification is therefore
When s and s' are both positive, as in figure 4.4a, the negative sign of the magnification signifies an inverted image. 4.6 IMAGE FORMATION When an object is placed on one side or the other of a converging lens and beyond the focal plane, an image is formed on the opposite side. If the object is moved closer to the primary focal plane, the image will be formed farther away from the secondary focal plane and will be larger, i.e., magnified. If the object is moved farther away from F, the image will be formed closer to F' and will be smaller. In figure 4.5 all the rays coming from an object point Q are shown as brought to a focus Q', and the rays from another point M are brought to a focus at M'. Such ideal conditions and the formulas given in this chapter hold only for paraxial rays, i.e., rays close to lens axis and making small angles with it.
- construction graphcally to determne state of image form in conves lens
Figure 4.5: Image formation by an ideal thin lens. All rays from an object point Q which pass through the lens are refracted to pass through the image point Q'.
4.7 LENS MAKERS' FORMULA If a lens is to be ground to some specified focal length, the refractive index of the glass must be known. Supposing the index to be known, the radii of curvature must be so chosen as to satisfy the equation
As the rays travel from left to right through a lens, all convex surfaces are taken as having a positive radius and all concave surfaces a negative radius. For an equiconvex lens , r 1 for the first surface is positive and r 2 for the second surface negative. Substituting the value of l/ f from equation(4.1), we can write
4.8 DERIVATION OF THE LENS FORMULA
A diagram for derivation the equation 4.1 ( lens formula ) is presented in figure 4.6 , which shows only two rays leading from the object of height y to the image of height y'. Let sand s' represent the object and image distances from the lens center and x and x' their respective distances from the focal points F and F'. From similar triangles Q'TS and F' TA the proportionality between corresponding sides gives
Another form of the lens formula is the Newtonian form, is obtained in an analogous way from two other sets of similar triangles, QMF and FAS on the one hand and TAF' and F'M'Q' on the other. We find
(4.10) Multiplication of one equation by the other gives
In the Gaussian formula the object distances are measured from the center of lens, while in the Newtonian formula they are measured from the focal points. Object distances ( s or x ) are positive if the object lies to the left of its reference point (A or F, respectively), while image distances (s' or x') are positive if the image lies to the right of its reference point (A or F', respectively). The lateral magnification as given by Eq. (4c) corresponds to the Gaussian form. When distances are measured from focal points, one should use the Newtonian form, which can be obtained directly from equation (4.10)
In the more general case where the medium on the two sides of the lens is different, it will be shown in the next section that the primary and secondary focal distances f and f' are different, being in the same ratio as the two refractive indices. The newtonian lens formula then takes the symmetrical form
The result is that the object and image must be on the opposite sides of their respective focal points.
4.9 DERIVATION OF THE LENS MAKERS' FORMULA
T he geometry required for this derivation is shown in figure 4.7. Let n, n', and n'' represent the refractive indices of the three media as shown, fl and
; the focal lengths for the first surface alone, and the focal lengths for the second surface alone.
Figure 4.7: Each surface of a thin lens has its own focal points and focal lengths, as well as separate object and image distances.
The oblique ray MT 1 is incident on the first surface as though it came from an axial object point M at a distance s 1 from the vertex A 1. At T 1 the ray is refracted according to Eq. (3b) and is directed toward the conjugate point M':
Arriving at T 2 , the same ray is refracted in the new direction T 2 M". For this second surface the object ray T 1 T 2 has for its object distance s 2 ', and the refracted ray gives an image distance of s 2 ''. When the following equation determined in chapter 3 is applied to second refracting surface,
In words, the focal lengths have the ratio of the refractive indices of the two media n and n" , see figure(4.8)
Figure 4.8 : When the media on the two sides of a thin lens have different indices, the primary and secondary focal lengths are not equal and the ray through the lens center is deviated.
If the medium on both sides is the same, n = n" , equation (4.15) reduces to
Note that the minus sign in the last factor arises when n" and n' are reversed for the removal of like terms in the last factor of equation (4.15). Finally, if the surrounding medium is air (n = 1), we obtain the lens makers' Formula
4.10 POWER OF LENS
The power of a lens is the measure of its ability to produce convergence of a parallel beam of light. A convex lens of large focal length produces a small converging effect of the rays and a convex lens of a small focal length produce a large converging effect. Concerning, the concave lens, it produces divergence. The power of convex and for concave takes +ve and – ve respectively. If the distances are measured in meter, the unit of power of lens is called diopter (D). the power of lens may be calculated by the relation
where P is the power of lens and f is the focal lens. Now, the power of lens by depending on the equation (4.19) can be written as
where
and are called reduced vengeance because they are direct measures of the convergence and divergence of the object and image wave front.
The divergent wave from the object s is positive, and so is its,. For convergent wave from the object s is negative , and so is its,.
The second step is to imagine the second lens in place and to make the following changes. Since ray 9 is seen to pass through the center of lens 2, it will emerge without deviation from its previous direction. Since ray 7 between the lenses is parallel to the axis, it will upon refraction by the
second lens pass through its secondary focal point_._ The intersection of
rays 9 and 11 locates the final image point. and are conjugate
points for the first lens, and are conjugate points for the second
lens, and and are conjugate for the combination of lenses. When the image is drawn in, corresponding pairs of conjugate points on the axis are and , and , and ,.
Figure 4.9: The parallel-ray method for graphically locating the final image formed by two thin lenses.
Problems
1: Two thin converging lenses of focal lengths ( f = + 3cm) and ( f= +4cm) respectively are placed in air and separated by a distance of 2 cm. an object is placed 4cm in front of the first lens. Find the position and the nature of the image and its lateral magnification. 2 : If an object is located 6.0 cm in front of a lens of focal length + 10. cm, where will the image be formed?
3 : An object is placed 12.0cm in front of a diverging lens of focal length 6.0 cm. Find the image. 4 : A plano-convex lens having focal length of 25.0 cm is to be made of glass of refractive index n = 1.520. Calculate the radius of curvature of the grinding and polishing tools that must be used to make this lens. 5 : The radii of both surfaces of an equiconvex lens of index 1.60 are equal to 8.0 cm. Find its power.
