Lens - Types, Formula and Power

 Lens

A portion of refracting material bound between two spherical surfaces is called a lens. 

Types of lens:

There are two types of lenses-

     (i) Converging lens (Convex lens). A lens is said to be converging if the width of a beam decreases after refraction through it. 

     Focal length of a converging lens is taken to be positive. Different types of converging lenses are shown below in Fig. 1.

Fig. 1. Converging lens. 

     Bi-convex lens. When radii of curvature of two sides of the convex lens is same then a double convex lens becomes a bi-convex lens. 

   (ii) Diverging lens (Concave lens). A lens is said to be diverging if the width of the beam increases after refraction through it. 

     Focal length of a diverging lens is taken as negative. Different types of diverging lenses are shown below in Fig. 2.

Fig. 2. Diverging lens. 

     Rule for nomenclature of a lens. While assigning a name to a lens, name of the surface having large radius of curvature is to be written first. 

Some definitions connected with lenses

     (i) Centre of curvature (C₁ and C₂). There are two centres of curvature for a lens, one each belonging to both the surfaces. 

     Centre of curvature of a surface of a lens is defined as the centre of that sphere of which that surface forms a part. 

    (ii) Radii of curvature (R₁ and R₂). There are two radii of curvature of a lens, one each belonging to both the surfaces. 

     Radius of curvature of a surface of a lens is defined as the radius of that sphere of which the surface forms a part. 

   (iii) Optical centre (C). It is a point, laying on the principal axis of the lens, with in it or outside such that a ray of light passing through it goes undeviated. 

     A ray of light AB which coincides with the principal axis goes undeviated and undisplaced after passing the optical centre C [Fig. 3]. 

Fig. 3. Refraction through a convex lens. 

     Any other ray A'CB', which also happens to pass through the optical centre C also goes undeviated but not undisplaced. In this case, the emergent ray EB' will be parallel to incident ray A'D. Position of the optical centre is so chosen that it divides the thickness of the lens in the direct ratio of the radii of curvature of the two surfaces. If the two surfaces are of same radius of curvature optical centre lies exactly in the center of the lens. 

   (iv) Principal axis. A line joining the two centres of curvature and passing through the optical centre is called principal axis. 

    (v) Principal foci. Two principal foci can be defined in case of a lens. 

     (a) First principal focus (F₁). It is a point situated on the principal axis of the lens such that a beam of light starting from it in case of a convex lens, [Fig. 4(i)] or directed towards it in case of a concave lens, [Fig. 4(ii)] becomes parallel to principal axis after refraction through the lens. 

Fig. 4. First principal focus. 

     (b) Second principal focus (F₂). It is a point situated on the principal axis of the lens such that a beam of light coming  parallel to principal axis meets in case of a convex lens, [Fig. 5(i)] or appears to meet in case of a concave lens, [Fig. 5(ii)] after refraction through the lens. 

Fig. 5. Second principal focus. 

    (vi) Focal lengths. Corresponding to the two principal foci, there are two focal lengths in case of lenses. 

     (a) First focal length. Distance of first principal focus from the optical centre of the lens is known as first principal focus length. It is denoted by f₁  [Fig. 4]. 

     (b) Second focal length. Distance of second principal focus from the optical centre of the lens is known as second focal length. It is denoted by f₂ [Fig. 5].

     For beams of light coming parallel to the principal axis, principal foci shall be on the point where principal axis cuts the focal plane. If the incident beam has rays parallel to themselves but not parallel to principal axis, the focus (not principal focus now) shall be on the focal plane, a certain distance below/above the principal axis. Various possible cases for a converging and diverging lens is shown in Fig. 6.

   (vii) Focal planes. Again corresponding to two principal foci, there are two focal planes. 

     (a) First focal plane (P₁). It is a plane passing through the first principal focus and perpendicular to principal axis as shown in [Fig. 6(i) and (ii)].

     (b) Second focal plane (P₂). It is a plane passing through the second principal focus and perpendicular to principal axis as shown in [Fig. 6(i) and (ii)].

Fig. 6. Position of focus. 

Action of lenses

Action of a lens (converging and diverging) can be explained by imagining the lens to be consisting of a number of prisms placed one above the other. 

     (i) Converging lens. A converging lens XY can be supposed to be consisting of a number of prisms placed one above the other while its central region can be treated as a glass slab [Fig. 7].

Fig. 7. Converging action of a convex lens. 

     Refracting edges of these imaginary prisms, in upper half, are directed upward while for those in lower half are directed downwards. Since a prism bends a ray towards the base rays crossing the upper half bend downwards while those crossing the lower half bend upwards. The rays passing through the central region go undeviated (imagined to be passing through a slab). The net effect on the beam is that the emergent beam meets at a point and is said to have converged. 

    (ii) Diverging lens. The refracting edges of the imaginary prisms of a diverging lens, in upper half are downwards while for those in lower half are upwards [Fig. 8].

Fig. 8. Diverging action of a concave lens. 

     Trying to bend the rays towards the base, prism bends the rays in upper half upward while those in lower half are bent downwards. As the result, the beam becomes more divergent and appears to meet at a point when produced backwards. Again in this case, the central region which is equivalent to a glass slab sends the rays undeviated. 

Lens formula (1/f = 1/v - 1/u) 

It is a relation connecting focal length of the lens with the distances of objects and images. 

     Sign conventions. Following sign conventions will be used for refraction through lenses. 

     (i) All the distances will be measured from the pole of the surface. 

    (ii) Distances measured against the incident ray are taken as negative while those measured along the incident ray are taken as positive. 

   (iii) All transverse measurements done above the principal axis are taken as positive while those done below the principal axis are taken as negative.

     Assumptions. All relations derived for refraction through lenses will hold good if the following conditions are satisfied. 

     (i) The source is point one. 

    (ii) The aperture of the lens is small. 

   (iii) Lens is thin. 

   (iv) Rays of light make smaller angles with principal axis. 

     Let 'AB' be an object situated on the principal axis of a lens. Consider two rays AM and AC. First goes parallel to the principal axis while second is heading towards the optical centre. If the object is situated beyond 'F' of a convex lens, a real image A₁B₁ is formed as shown in [Fig. 9(i)]. If the object is situated in between 'C' and 'F' of a convex lens, a virtual image  A₁B₁ is formed as shown in [Fig. 9(ii)]. In case of a concave lens also, a virtual image A₁B₁ is formed as shown in [Fig. 9(iii)].

Fig. 9. Refraction through a lens. 

     Triangles ABC and A₁B₁C₁ are similar, 

     ∴            AB/A₁B₁ = BC/B₁C                    ... (1) 

     Triangles CMF and A₁B₁F are also similar, 

     ∴          CM/A₁B₁ = CF/B₁F

     Since           CM = AB

     ∴          AB/A₁B₁ = CF/B₁F                     ... (2) 

     From equations (1) and (2), we get

                 BC/B₁C = CF/B₁F                       ... (3) 

     Case (i) Convex lens producing a real image

     From [Fig. 9(i)]. B₁F = B₁C - CF

     Substituting for B₁F in equation (3), we get

               BC/B₁C = CF/B₁C - CF                ... (4) 

     According to sign convention, 

             BC = -u, B₁C = +v, CF = +f

     Substituting these un equation (4), we get

                   -u/v = f/v - f

or              -u (v - f) = vf

or             -uv + uf = vf

or              uf - vf = uv

     Dividing throughout by uvf, we get

            uf/uvf - vf/uvf = uv/uvf

                                1/f = 1/v - 1/u

     Case (ii) Convex lens producing a virtual image

     From [Fig. 9(ii)], B₁F = B₁C + CF

     Substituting for B₁F in equation (3), we get

            BC/B₁C = CF/B₁C + CF                     ... (5) 

     According to sign convention, 

             BC = -u, B₁C = -v, CF = +f

     Substituting these in equation (5), we get

              -u/-v = f/-v + f

       -u (-v + f) = -vf, +uv - uf = -vf, uf - vf = uv

     Dividing throughout by uvf, we get

            uf/uvf - vf/uvf = uv/uvf

or                            1/f = 1/v - 1/u

     Case (iii) Concave lens producing a virtual image

     From [Fig. 9(iii)], B₁F = CF - B₁C

     Substituting for B₁F in equation (3), we get

               BC/B₁C = CF/CF - B₁C                   ... (6) 

     According to sign convention, 

            BC = -u, B₁C = -v, CF = -f

     Substituting these in equation (6), we get

                -u/-v = -f/-f - (-v) 

or           -u/-v = -f/-f +v

or          -u (-f + v) = vf

or              +if - uv = vf

or                uf - vf = uv

      Dividing throughout by uvf, we get

              uf/uvf - vf/uvf = uv/uvf

or                             1/f = 1/v - 1/u

     Thus, it is clear that whatever the type of lena may be and the image produced by it may be real or virtual, u, v and f are connected together by the relation

                1/f = 1/v - 1/u                               ... (7) 

     Equation (7) is called lens formula. 

Linear magnification

It is the ratio between the size of the image to the size of the object. 

     Let the size of the image and the size of the object be denoted by 'I' and 'O' respectively. 

                      m = I/O                                 ... (8) 

(a) Magnification produced by a convex lens

     (i) Real image

     Triangles ABC and A₁B₁C [Fig. 9(i)] are similar, 

                 A₁B₁/AB = CB₁/CB

     According to sign convention, 

         A₁B₁ = -I   (Measurement below principal axis) 

          AB = +O   (Measurement above principal axis) 

         CB₁ = +v   (Along incident ray) 

          CB = -u   (Against incident ray) 

     Making the substitution

              -I/O = +v/-u

     ∴       I/O = v/u                                      ... (9) 

     Using equation (8) and (9), 

                 m = v/u

    (ii) Virtual image

     Triangles ABC and A₁B₁C [in Fig. 9(ii)] are similar

            A₁B₁/AB = CB₁/CB

     Here

           A₁B₁ = +I     (Measurement above the principal axis) 

             AB = +O    (Measurement above the principal axis) 

            CB₁ = -v     (Against incident ray) 

            CB = -u      (Against incident ray) 

     Making substitution, 

               +I/+O = -v/-u

or               I/O = v/u                                 ... (10) 

     From equation (8) and (9) 

                     m = v/u

(b) Magnification produced by a concave lens

     Triangles A₁B₁C and ABC [in Fig. 9(iii)] are similar

     ∴          A₁B₁/AB = CB₁/CB

     Applying sign conventions, 

         A₁B₁ = +I    (Measurement above the principal axis) 

           AB = +O   (Measurement above the principal axis) 

          CB₁ = -v     (Against incident ray) 

           CB = -u     (Against incident ray) 

     Making the substitution

              +I/+O = -v/-u

or              I/O = v/u                                   ... (11) 

     Using equation (8) and (9), 

                    m = v/u

     Thus, it is clear that the linear magnification produced by a lens is 

                     m = v/u

     This relation true both for convex and concave lenses. This relation is also independent of the nature of image (real or virtual). 

Expression for m in terms of u, v and f

     (i) In terms of v and f

     For a lens,         1/f = 1/v - 1/u

     Multiplying throughout by v, 

                  v/f = 1- (v/u) = 1 - m

     ∴             m = 1 - (v/f) 

     ∴             m = f - v/f

    (ii) In terms of u and f

     Again, for a lens     1/f = 1/v - 1/u

     Multiplying throughout by u, 

               u/f = u/v - (1) = 1/m - (1) 

     ∴        1/m = 1 + (u/f) = f + u/f

     ∴           m = f/f + u

     Thus, linear magnification for a lens can be expressed as

               m = I/O = v/u = f + v/f = f/f + u

Position of the image as the objective is gradually moved from infinity to the pole of the lens

As the object is shifted gradually from infinity to the pole of the lens, the image produced by the lens undergoes a variation not only in nature and size but its position is also affected. We discuss below the different cases of relative change in position, nature and size of the image as the object is shifted. 

(i) Object being at infinity

     In this case,          u = ∞

     ∴                    1/f = 1/v - 1/∞

or                       1/f = 1/v    or      v = f

     Therefore, the image 'A₁B₁' is obtained at a distance 'f' from the lens, i.e., on the principal focus 'F' of the lens. Formation of the image by a convex lens is shown in (Fig. 10). Magnification in this case is extremely small and the image is said to be real and inverted. 

Fig. 10. Object at infinity. 

(ii) Object lying beyond '2f'

     According to lens formula,  

                        1/f = 1/v - 1/u

     ∴                1/v = 1/f + 1/u

     If u > - 2f, 2f > v > -f, i.e., the image lies between f and 2f, m(= v/u) always is less than one. 

     Thus, as object moves from infinity to the position 2f, position of image changes from 'f' to '2f' distance away from lens. Formation of image is shown in (Fig. 11). The image is diminished in size and is real and inverted. 

Fig. 11. Object beyond 2f.

(iii) Object at 2f [Fig. 12]

                        u = -2f

     ∴               1/f = 1/v - (1/-2f) = 1/v + 1/2f

or                  1/v = 1/f - 1/2f = 1/2f

     ∴                  v = 2f

     ∴                 m = v/u = 2f/-2f = -1

Fig. 12. Object at 2f.

     Thus, the image of an object situated at '2f' is also obtained at the same distance on the other side of the lens. The formation of the image is shown in (Fig. 12). The size of the image is same as that of the object and is real and inverted. 

(iv) Object lying between 'f' and '2f'

     According to lens formula, 

                          1/f = 1/v - 1/u

     ∴                  1/v = 1/f + 1/u

     If -2f > u > -f, i.e., can be senn from above relation that, v > 2f.

     Thus, the image lies at a distance greater than 2f on the other side of the lens, 

     ∴    m (= v/u) is greater than one. 

Fig. 13. Object between f and 2f.

     Thus, if an object is moved from '2f' to 'f' the image goes beyond '2f'. Formation of image is shown in (Fig. 13). The image is magnified in size and is real and inverted. 

(v) Object at f

     In this case,         u = -f

     ∴                          1/f = 1/v - 1/u

     ∴                         1/v = 1/f + 1/u = 1/f - (1/-f) 

     ∴                            v = ∞

     Since u = -f and v = ∞

     ∴    m (= v/u) is infinite. 

Fig. 14. Object at f. 

     Therefore, the image of an object situated at a distance 'f' from the lens, i.e., on the principal focus is obtained at infinity. The formation of the image is shown in (Fig. 14). The image is infinitely large in size and is real and inverted. 

(vi) Object lying between 'f' and optical centre 'C'

     At f,              u = -f

     ∴                 1/f = 1/v - (1/-f) = 1/v + 1/f

or                   1/v = 1/f - 1/f = 0

     ∴                   v = ∞

     At C,            u = 0.

     ∴               1/v = 1/f + 1/0 = ∞     

     ∴                  v = ∞

     Thus, as the object is moved from principal focus 'F' to the optical centre 'C' of the lens, its image moves from infinity to the optical centre and lies on the same side of the lens. Formation of the image is shown in (Fig. 15). The image is magnified, virtual and erect. 

Fig. 15. Object between F and optical center. 

     

     Conclusion. A change in nature, size and position of the image due to a change in the position of object can be summed up in the form of following table. 

S. No.	Position of object	Position of image	Relative size of image	Nature of image
1.	At infinity	At f	Very small	Real, inverted
2.	Beyond 2f	Between f and 2f	Diminished	Real, inverted
3.	At 2f	At 2f	Same size	Real, inverted
4.	Between f and 2f	Beyond 2f	Large	Real, inverted
5.	At f	At infinity	Very large	Real, inverted
6.	Between f and pole	On same side as the object, away from lens	Large	Virtual, erect

Power of a lens

The function of lens is to converge a beam of light (in the case of convex lens) or to diverge it (in the case of concave lens). Since a convex lens of shorter focal length produces a greater convergence, therefore, power of a lens varies inversely as its focal length. 

     The reciprocal of the focal length of a lens, expressed in metre, is called its power. 

     Representing focal length of a lens by 'f' in metre and power by 'P', we can write

                    P = 1/f(m) 

    The unit of power is dioptre is defined as the power of a lens of focal length one metre. 

     Thus,     P (dioptre) = 1/f(metre) 

                                        = 100/f(cm) 

     Thus, the power of a convex lens of focal length 25 cm is 4 dioptre. Opticians specify lenses by their powers. The power of a convex lens is taken as positive and that of a concave lens negative. 

Key formulae

1. Lens formula : 1/f = 1/v - 1/u

2. Magnification (linear) : 

            m = I/O = v/u = f - v/f = f/f + u

3. Lens maker's formula :

     (i) When the lens (μ₂)  has same medium (μ₁) on its two sides

               1/f = (1μ₂ - 1) (1/R1 - 1/R2)

    (ii) When medium situated on the two sides of the lens is different. 

             1/f = μ - μ1/μ3R1 + μ3 - μ₂/μ3R2 

4. Power of a lens :

             P (dioptre) = 1/f(metre) = 100/f(cm) 

Refraction, its law and optical fibre

Refraction:

Velocity of light in different media is different. As a Ray of light goes from medium 1 (velocity of light v₁) to medium 2 (velocity of light v₂), its velocity for changes on crossing the interface XY. The phenomenon is called refraction.

Refraction is the phenomenon by virtue of which a ray of light going from one medium to the other undergoes a change in its velocity. 

If  the ray is incident, obliquely as shown in Fig.1, 

Fig. 1. Refraction at a plane surface

the change in velocity of light results in a change in its path also while there is no change in path if the ray is incident normally. The ray which approaches the interface is called incident ray (SO). Ray which goes into the second medium is called refracted ray. Angle between the incident ray and the normal to the interface at the point of incidence is called angle of incidence (i). Angle between the refracted ray and the normal to the interface is called the angle of refraction (r). 

If light travels from an optically rarer medium to a denser medium, it bends towards the normal, i.e., r is less than i. If it travels from denser to rarer medium, it bends away from the normal, i.e., r is greater than i. 

Important Notes

During refraction the path of light may or may not change. 

1. For  beam of light incident normally, there is no change of path. 
2. For a beam going from rarer to denser medium, the beam bends towards the normal. 
3. For a beam going from denser to rarer medium, the beam bends away from the normal. 

Law of Refraction:

Phenomenon of refraction is governed by the following two laws, called laws of refraction:

(i) The sine of the angle of incidence bears a constant ratio with the sine of the angle of refraction. 

i.e.,       sin i/sin r = constant

The law is often termed as Snell's law. 

(ii) The incident ray, the refracted ray and the normal to the interface at the point of incidence all lie in one plane and that plane is perpendicular to the interface separating the two media. 

Refractive Index:

Refractive index of a medium is a characteristic of medium which determines its behaviour to propagation of light. It is sometimes, taken to be a measure of the optical density of the medium. A medium having a greater value of refractive index is said to be a optically denser then that having a lower value. Refractive index of vacuum if the smallest value and is equal to one. All other medii have refractive indices greater than one when measured with respect to vacuum. In such a case in refractive index of the medium is termed as absolute refractive index. It can be defined as a number of ways. 

(i) Definition in terms of angles of incidence and refraction

     Let XY be the interface separating the two media 1 and 2 from each other [in Fig.1]. Let i and r be the angle of incidence and refraction respectively. 

     According to Snell's law, 

     sin i/sin r = constant = ¹μ₂                  ... (1) 

where ¹μ₂ is called the refractive index of second medium with respect to first. It is always referred to as the refractive index of that medium to which the light goes with respect to the medium from which it comes. 

     Refractive index of a medium with respect to another is defined as the ratio between sine of the angle of incidence to the sine of angle of refraction. 

(ii) Definition in terms of velocity of light

     Refractive index of medium 2 with respect to 1 is also defined as the ratio between velocity of light in medium 1 to the velocity of light in medium 2.

     If v₁ and v₂ are the velocities of light in first and second medium, 

                ¹μ₂ = v₁/v₂                                   .... (2) 

     If the first medium is air or vacuum, the refractive index is written as ⁰μ₂ or simply as μ and is known as the absolute refractive index. 

                 μ = c/v                                        .... (3) 

     where 'c' is the velocity of light in vacuum and 'v' is the velocity of light in that medium. 

(iii) Definition in terms of wavelength of light

     As light goes from one medium to another, there is no change in its frequency. Since v = nλ, a change if medium shall result in the change in wavelength of light. 

              v₁ = nλ₁,            v₂ = nλ₂

  Substituting for  v₁ and   v₂ in equation (2), 

           ¹μ₂ = nλ₁/nλ₂ = λ₁/λ₂                       ... (4) 

     Refractive index of second medium with respect to first is defined as the ratio between wave-length of light in medium 1 to the wavelength of light in medium 2.

(iv) Definition in terms of absolute refractive indices of the medium

     Dividing the numerator and denominator of equation (2) by c, we get

¹μ₂ = v₁/c / v₂/c          [μ₁ = c/v₁      ∴  1/μ₁ = v₁/c]

      = 1/μ₁ / 1/μ₂         [μ₂ = c/v₂      ∴  1/μ₂ = v₂/c]

  or       ¹μ₂ = μ₂/μ₁                                     .... (5) 

     Equation (5) provides another definition of refractive index. 

     Refractive index of second medium with respect to first is defined as the ratio between absolute refractive index of second medium to the absolute refractive index of first medium. 

Relative Magnitudes of i and r

Relative magnitudes of i and r, during refraction, depend upon the nature of two media across an interface. Let XY be an interface separating the media 1 and 2 from each other such that

Case (i) Light travelling from rarer to denser medium [Fig.2(i)].

     It shall be observed that the ray, on entering the second medium bends towards the normal. So, in this case r is less than i. Angle of deviation = (i - r) 

Fig. 2. Bending of light due to refraction

Case (ii) Light travelling from denser to rarer medium [Fig.2(ii)].

    

      The ray in such a case shall bend away from the normal thus making r greater than i. Angle of deviation = (r - i). 

Principle of reversibility

It states that if a ray of light, after suffering a number of reflections and/or refractions has its path reversed at any stage, it will retrace itself back to the source along the same path. 

Fig. 3. Principle of reversibility. 

     Consider a ray SO, travelling in medium 2 striking the interface XY [Fig. 3]. The ray changes its path after refraction. If the path of the ray is reversed by placing a plane mirror normal to its path, the ray goes from medium 2 to medium 1 and reaches the source S. 

     Considering journey of ray from medium 1 to 2 

               ¹μ₂ = sin i/sin r                            ... (6) 

     As the ray goes from medium 2 to medium 1, the angle of incidence and the refraction get their roles interchanged. Therefore, while going from medium 2 to medium 1.

               ²μ₁ = sin r/sin i                          ... (7) 

     Multiplying equations (6) and (7), we get

            ¹μ₂ × ²μ₁   = sin i/sin r × sin r/sin i = 1,            

             ¹μ₂ = 1/²μ₁

     For example, 

              airμglass = aμg = 1.5

     ∴      glassμair  = gμa = 1.5

Important notes 

     Velocity of light 'c' is maximum in vaccum. Since refractive index μ = c/v, there is no existence of a medium having μ < l. 

Optical Fibre

According to the property of rectilinear propagation of light, light travels in a straight line. Making use of the phenomenon of total internal reflection, it can be made to go in a curved path. Consider, a curved rod AB [Fig. 4(i)] made up of a transparent material like glass or optical grade plastic. A ray of light entering the pipe through face A undergoes successive total internal reflections at points P₁, P₂, .... and emerges out of face B. AB is called a light pipe in the sense that it has allowed light to flow through it from end A to B just like a hollow pipe allowing flow of water through it. We can use this pipe in the case where the order of transmission of different parts of an image is not important, but we require only a beam of light. 

Fig. 4. Optical fibre. 

     An optical fibre consists of a glass or a plastic core surrounding by a cladding made up of a similar material but with a lower refractive index [Fig. 4(ii)]. Light propagates through an optical fibre in two modes. 

     (i) Monomode propagation. In this case light has only one propagation path along the length of the core. As light propagates, cladding causes total internal reflection. 

    (ii) Multimode propagation. There are many propagation paths while the reflection takes place from the edges of the core. The core in this case, is of greater diameter. Due to pulse dispersion in this case, the data is transmitted at a lesser rate than that in monomode propagation. 

     There are three types of optical fibres which are in common use. 

     (i) Stepped index multimode optical fibre. The core and cladding, both are of uniform refractive index. Thus, there is an abrupt change in refractive index at the interface. The diameter of the core is about 50-60 μm while that of cladding is 125 μm. Because of high pulse dispersion bandwidth is limited between 10 MHz km to 50 MHz km. 

    (ii) Stepped index monomode optical fibre. The core and cladding both are again of uniform refractive index. Core diameter is small, 1-10 μm and cladding diameter is 125 μm. Due to smaller pulse dispersion in it, bandwidth of several GHz km is attainable. 

   (iii) Graded index multimode optical fibre. The core consists of a material having non uniform refractive index being maximum at the centre and minimum at the interface. The distribution of refractive index is parabolic. The core diameter is 50-60 μm while cladding diameter is 125 μm. In this case, velocity of propagation is higher than that in 'stepped index' multimode. Therefore, there is lesser pulse dispersion with a consequent increase in bandwidth up to 1 GHz km. 

     Following comparison of transmission through optical fibre and through electrical cable shall illustrate the superiority of optical fibre. 

Optical fibre

Electrical cable

1. Data communicated at higher speed. 2. Data transmission rate is 1 G bit s−1.  3. No electromagnetic interferences, lightning strikes etc.  4. No cross talks or reflection problems.  5. Attenuation increases with distance at a lesser rate.  6. Bit error one in 10⁹.  7. No risk of short circuits or electrical spark.  8. Lighter in weight.  9. Tempering of data not easy.  10. More resistive to corrosive atmosphere.  11. Installation and maintenance over long distances less costly.  12. Greater operational life.

1. Data communicated at comparatively lower speed.  2. Data transmission rate for (i) twisted pair - 100 M bit s−1.  (ii) Base band transmission in co-axial cable - 30 M bit s−1.  (iii) board band r.f. transmission in co-axial cable - 400 M bit s−1.  3. Electromagnetic interferences and lightning strikes effective.  4. Cross talks and reflection problem effective.  5. Attenuation increases distance at a large rate.  6. Bit error one in 10⁶.  7. Risk of short circuit and electrical spark.  8. Heavier in weight.  9. Tempering of data easy.  10. Prone to corrosive atmosphere.  11. Installation and maintenance over long distances more costly.  12. Lesser operational life.

Key Formulae

1. Law of refraction :

     Snell's law,     sin i/sin r =constant

2. Refractive index :

        ¹μ₂ sin i/sin r = v₁/v₂ = μ₂/μ₁

     Absolute refractive index,

         μ = real depth/apparent depth

3. Principle of reversibility :

          ¹μ₂ × ²μ₁ = 1

4. Lateral shift through a glass slab :

         d = t sec r sin (i - r) 

5. Critical angle : 

         sin C = 1/μ

6. Refraction at a single surface :

     (i) Light going from rarer to denser medium

     (General formula) 

          μ₂/v - μ₁/u = μ₂ - μ₁/R

     (Air to Glass) 

          μ/v - 1/u = μ - 1/R

    (ii) Light going from denser to rarer medium

      (General formula) 

          μ₂/u - μ₁/v = μ₂ - μ₁/R

     (Glass to Air) 

          μ/u - 1/v = μ - 1/R

Mirror Formula (1/f = 1/v + 1/u = 2/R)

 Spherical Mirror

1. Spherical mirror. It is a polished surface which forms the part of a sphere. It is of two types :

     (a) Concave mirror. It is a spherical mirror which when looked from the reflecting side is depressed at the centre and bulging at the edges. It is shown in [Fig. 1(i)].

Fig. 1. Characteristics of a spherical mirror. 

     (b) Convex mirror. It is a spherical mirror which when looked from the reflecting side bulges at the centre and is depressed at the edges. It is shown in [Fig. 1(ii)].

2. Pole (P). The central point which is most depressed (in a concave mirror) and which is most bulging (in a convex mirror) is called pole of the mirror. 

3. Radius of curvature (R). Radius of curvature of a mirror is defined as the radius of that sphere of which the mirror forms a part. 

4. Centre of curvature (C). It is the centre of that sphere of which the surface forms a part. 

5. Principal axis. A straight line joining the pole of the mirror and its centre of curvature is called principal axis. 

6. Aperture (XY). The diameter of the circular outline of the mirror is called aperture of the mirror. 

7. Principal section. A section of the mirror by a plane passing through the centre of curvature and the pole is called principal section. 

8. Principal focus. Consider a beam of light coming parallel to principal axis and incident on a spherical mirror. In case if a concave mirror [Fig. 2(i)], the beam actually meets at F while in the case of convex mirror it only appears to diverge from F, when the rays are produced back [Fig. 2(ii)]. They meet at the point 'F' and is called principal focus. 

Fig. 2. Principal focus of a spherical mirror. 

     Principal focus is a point situated on the principal axis at which a beam coming parallel to principal axis meets or appears to meet after reflection from the mirror. 

9. Focal plane. It is a vertical plane passing through the principal focus and perpendicular to the principal axis. 

10. Focal length (f). Focal length of a spherical mirror is the distance of its principal focus from its pole. It is denoted by 'f' as shown in [Fig. 2(i) and (ii)]. Focal length is measured in terms of units of length i.e. cm or m. 

Sign Conventions

While handling derivations in optics we have to deal with measurement of certain distances like distance of object (u), distance of image (v), focal length (f) and radius of curvature (R) etc. We shall adopt the following sign conventions for their measurements :

     (i) All the ray diagrams will be drawn with light travelling from left to right. 

    (ii) The pole of the surface (in case of lenses) will be considered to be situated at the origin of a system of co-ordinate axes. All the measurements will be done from the origin. 

  (iii) The principal axis will always coincide with X-axis of the system of co-ordinate axis. 

   (iv) All the distances measured towards left of origin are taken as negative while, those measured towards right of origin are taken as positive. 

   (v) All transverse measurements done above the X-axis are taken as positive while those done below the X-axis are taken as negative. 

Assumption

While obtaining some relations, in ray optics, we shall make some assumptions given as follows. All those formulae will hold good only if these conditions are satisfied. 

     (i) The sources are considered to be point one. 

    (ii) The aperture of the surface/lens should be small. 

   (iii) Rays of light should make smaller angles with the principal axis. 

Relation between focal length and radius of curvature

Considered a ray of light coming from infinity (parallel to the principal axis), incident on a spherical mirror at A. If the mirror is concave [Fig. 3(i)], it meets at F after reflection. If the mirror is convex [Fig. 3(ii)], it appears to come from F after reflection. The angle of incidence and angle of reflection are shown in the diagram. 

     (i) Concave mirror

                ㄥSAC = ㄥACF = ㄥi   [alternate angles]

               ㄥCAF = ㄥr

     According to law of reflection

              ㄥi = ㄥr      ∴    ㄥCAF = ㄥFCA

    (ii) Convex mirror

              ㄥSAN = ㄥPCA = i  [corresponding angles]

             ㄥNAI = ㄥFAC = r  [vertically opposite angles]

     According to law of reflection

                      ㄥi = ㄥr

     ∴      ㄥCAF = ㄥFCA

     ∴        ㄥAF = FC 

     From [Fig. 3 (i) and (ii)], radius of curvature R is 

     R = PC = PF + FC = PF + AF = 2PF     ... (i) 

     [For very small aperture AF ≃ FP]

     According to sign convention for concave mirror [Fig. 3(i)]

Fig. 3. Reflection at a spherical mirror. 

     CP ≃ -R, PF = -f   or   -R = -2f   or  f = R/2

     According to sign convention for convex mirror [Fig. 3(ii)]

             CP = + R, PF = + f

     Substituting in (i), 

               R = 2f  or   f = R/2

     Thus, the focal length of a spherical mirror is half its radius of curvature. 

Mirror Formula (1/f = 1/v + 1/u = 2/R) 

It is a relationship connecting object distance, image distance and focal length of a spherical mirror. 

     Consider a spherical mirror having principal section XPY and focus F. AB is an object standing vertically on the principal axis at a point B. (In case of virtual image by concave mirror, the point B is within focus F). A ray AK parallel to principal axis after reflection from mirror either coverages at F (in case of concave) or appears to diverge from F (in case of convex). The ray passes along KR. Another ray AP is reflected from pole (P) and proceeds along PS. In [Fig. 4(i)], the two reflected rays KR and PS actually meet at a point A' giving the real image of a point A. In [Fig. 4(ii) and (iii)], the ray KR and PS are divergent forward apparently originating from point A' behind the mirrors, giving virtual image of the point A. Another ray BP from foot of object is incident normally and after reflection from mirror retraces the path (along PB). Draw a perpendicular from A' upon the principal axis. 

Fig. 4. Formation of image by a spherical mirror. 

     In [Fig. 4(i)], A'B' represents real and inverted image of AB. In [Fig. 4(ii)], A'B' represents virtual, erect and magnified image of AB. In [Fig. 4(iii)], A'B' represents virtual, erect and diminished image of AB. 

Deduction

     (i) Concave mirror producing real image [Fig. 4(i)]

        BP = distance of object from pole = u

       B'P = distance of image from pole = v

        PF = focal length = f

     

     ∆APB and ∆A'PB' are similar. 

     ∴         A'B'/AB = P'B/PB                       ... (ii) 

     Again for very small aperture KP is nearly straight. 

     ∆KPF, ∆A'B'F are similar. 

     ∴         A'B'/KP = B'F/PF

     ⇒       A'B'/AB = PB' - PF/PF    [∵  KP = AB]

     Using equation (ii), we get

             PB' - PF/ PF = PB' - PF/PF

     According to sign convention

            PB' = -v, PB = -u and PF = -f

            -v/-u = -v - (-f)/-f

     ⇒   vf = uv - uf      ⇒    uf + vf = uv

     Dividing by uvf, we get

            1/v +1/u = 1/f.

     (ii) Concave mirror producing virtual image [Fig. 4(ii)]

     ∆APB and ∆A'PB' are similar. 

     ∴         A'B'/AB = PB'/PB                      ... (iii) 

     Again ∆KPF and ∆A'B'F are similar. (KP is nearly straight for small aperture) 

     ∴         A'B'/KP = B'F/PF

     ⇒       A'B'/AB = PB' + PF/PF

     Using equation (iii), we get

             PB'/PB = PB' + PF/PF

     According to sign convention

             PB' = +v, PB = -u and PF = -f. 

     Making these substitutions

                   +v/-u = v - f/-f

or         -vf = -uv + uf       ⇒    uf + vf = uv

     Dividing by uvf, we get

           1/v + 1/u = 1/f.

     (iii) Convex mirror [Fig. 4(iii)]

     ∆APB and ∆A'PB' are similar. 

     ∴           A'B'/AB = PB'/PB

     Again ∆KPF, ∆A'B'F are similar. 

     ∴           A'B'/KP = B'F/PF

     (KP is straight for very small aperture) 

     ⇒         A'B' = PF - PB'/PF

     According to sign convention

             PB' = +v, PB = -u and PF = +f

     Making these substitutions

            v/-u = f - v/f    or    vf = -uf + uv

     ⇒   uf + vf = uv

     Dividing by uvf, we get

     ⇒       1/v + 1/u = 1/f.                            ... (iv) 

Magnification

Magnification gives us an idea about the relative change in size of image as compared to that of object. There are two types of magnification. 

1. Linear magnification or Transverse magnification

     Linear magnification produced by a spherical mirror is defined as the ratio between size of the image to the size of the object. 

     Let the size of the images and object be denoted by I and O respectively. The linear magnification produced by the spherical mirror is given by

                    m = I/O                                   ... (v) 

     Case (a). Magnification produced by a concave mirror

     ∆APB, ∆A'PB' [in Fig. 4(i)] are similar. 

     ∴        A'B'/AB = PB'/PB

     Using sign conventions, 

     A'B' = - I    (Measurement below the principal axis) 

     AB = +O    (Measurement above the principal axis) 

     PB' = -v     (Against incident ray) 

     PB = -u      (Against incident ray) 

     Making the substitution, 

           - I/O = -v/-u

or         I/O = -v/u                                      ... (vi) 

     From equations (v) and (vi), 

                 m = I/O = - (v/u)                      ... (vii)

     Case (b). Magnification produced by a convex mirror

     ∆APB and ∆A'PB' [in Fig. 4(iii)] are similar. 

     ∴         A'B'/AB = PB'/PB

     Using sign conventions, 

     A'B' = +I     (Measurement above the principal axis) 

     AB = +O     (Measurement above the principal axis) 

     PB' = +v     (Along incident ray) 

     PB = -u      (Against incident ray) 

     Making the substitution, 

           I/O = -v/u                                        ... (viii) 

     From equations (v) and (viii), we get,

           m = I/O = -v/u                                ... (ix) 

     It is clear from equations (vii) and (ix) that

     (i) Expression for linear magnification produced by a concave mirror and by a convex mirror is same. 

    (ii) Expression for magnification is independent of the nature of image (real/virtual). 

Relation between m and f

     (i) In term of v

     According to mirror formula, 

  

             1/v + 1/u = 1/f

     Multiplying throughout by v, 

             1 + v/u = v/f

     Since v/u = -m [from equation (ix)]

     ∴       1 - m = v/f      or      m = 1 - v/f

or                m = f - v/f                             ... (x) 

     (ii) In term of u

     Again, according to mirror formula

              1/v + 1/u = 1/f

     Multiplying throughout by u, 

             u/v + 1 = u/f

     Since u/v = - 1/m [from equation (ix)]

              -1/m + 1 = u/f

or                 1/m = 1 - u/f = f - u/f

     

     ∴                 m = f/f - u                             ... (xi) 

     Combining equations (ix), (x) and (xi), 

             m = I/O = -v/u = f - v/f = f/f - u

2. Longitudinal magnification (m') 

     When the object and the image possess finite size along the principal axis, it is desirable to compare them with each other. This is expressed in terms of longitudinal magnification. 

     Longitudinal magnification of a mirror is defined as the size of the image to the size of the object both measured along the direction of principal axis. 

     Let 'dv' and 'du' respectively be the distances occupied by the image and the object along principal axis. Longitudinal magnification m' is given by

                      m' = dv/du

     According to mirror formula, 

                    1/v + 1/u = 1/f

     Differentiate both side with respect to v. 

       - (1/v²) dv - (1/u²) du = 0    [∵  f = constant]

     ∴       m' = dv/du = - (v²/u²) = - [- (v/u)²]

     Since     - (v/u) = m   (transverse magnification) 

     ∴       m' = - m²

     or    Longitudinal magnification = - (transverse magnification) 

Relative positions, Size and Nature of images as object is brought from infinity to the pole of a concave mirror

As object is brought from infinity towards the pole of a concave mirror u changes and hence v also changes giving different position of images. 

      

(i) If the object is at infinity, u = ∞ (Fig. 5)

     

Fig. 5. Object at infinity. 

     For concave mirror, focal length = - f

   

     From mirror formula

                   1/v + 1/u = 1/f

     We get, 1/v + 1/∞ = 1/-f

     ⇒                     1/v = - (1/f). 

     Thus, image is obtained at the focus of focal plane. Linear magnification, 

               m = v/u = -f/∞ = 0

     Thus, image us very small in dimensions. Here rays from infinity are rendered into beam parallel to principal axis. After reflection rays coverage at focus giving real and point image at focus. 

    (ii) The object lies beyond centre of curvature (Fig. 6).

Fig. 6. Object beyond 2f.

i.e.,             ∞ > u > 2f

                   0 < 1/u < 1/2f

                   1/v + 1/u = 1/-f

     ⇒                    1/v = 1/-f + 1/u

     Since         u > 2f

     ⇒              1/v > - 1/f + 1/2f

                       1/v > - 1/2f    ⇒   v < - 2f

     Since          u < ∞

     Again        1/v < - 1/f - 0    ⇒     v > f. 

     Magnification,      m = (v/u) < 1

     Therefore, if        ∞ > u > 2f

                                    2f > u > f. 

     Thus, a real inverted and diminished image is formed in between focus and centre of curvature. 

   (iii) Object is at centre of curvature (Fig. 7) 

Fig. 7. Object at 2f.

i.e.,                 u = - 2f = - r

     ∴     1/v + 1/-2f = 1/-f

     ⇒                   v = - 2f.

     Magnification, 

                 m = v/u = - 2f/- 2f = 1.

     Thus, a real and inverted image of same size as that of object is formed at centre of curvature. 

    (iv) Object is in between a distance f and 2f, i.e., in between focus and centre of curvature (Fig. 8).

Fig. 8. Object between f and 2f.

     Here,          f < u < 2f; i.e., 1/f > 1/u > 1/2f

     Since, for     u = - f,  1/v = 1/- f + 1/u

                           1/v > 1/- f + 1/f   ⇒    1/v > 0

     ⇒                   v < 1/0 ; i.e.,  v < ∞

     Again, for      u = - 2f

                             1/v < 1/- f + 1/2f     

   

     ⇒                     1/v < - (1/2f) 

     ∴                      v > - 2f,    m = (v/u) > 1.

     Thus, a real inverted and magnified image is formed in between centre of curvature and infinity. 

     (v) Object is kept at focus (Fig. 9) 

Fig. 9. Object at f. 

i.e.,                         u = - f

                 1/v + 1/- f = 1/- f

     ∴                     1/v = 0

                               v = ∞

     The rays after reflection are rendered into a parallel beam meeting in infinity. 

    (vi) Object is kept within focus (Fig. 10) 

Fig. 10. Object between F and pole. 

i.e.,               u < - f

     ⇒           1/u > - (1/f) 

                    1/v = 1/- f + 1/u

     Since    -1/u > 1/f

     ∴   v is positive. 

     Thus, a virtual, erect and magnified image is formed on the other side of mirror. 

     All the cases discussed above are being put in a tabular form below :

S. No.	Position of object	Position of image	Size of image	Nature of image
1.	At infinity	At F	Very small	Real, inverted
2.	Beyond C	Between F and C	Diminished	Real, inverted
3.	At ac	At C	Equal in size	Real, inverted
4.	Between F and C	Beyond C	Magnified	Real, inverted
5.	At F	At infinity	Highly magnified	Real, inverted
6.	Between F and C	Behind the mirror	Magnified	Virtual, erect

Use of Spherical Mirror

 Spherical mirrors (concave and convex) have been put to use in daily life, in a number of ways. 

Use of Concave mirror

     1. When object is situated in between principal focus and pole of a concave mirror, through the image is virtual, it is highly magnified. As such a concave mirror can be used as a shaving mirror or make-up mirror. 

     2. Doctors use concave mirrors for focussing light on ear, nose, throat for their close examination. There is a hole at the centre of such a mirror. They wear it on their heads with the help of a belt in such a way that the hole comes directly in front of their eye. They can look through the hole. 

     3. Concave mirrors, when used as reflector, can be employed for constructing reflecting telescopes. 

Use of Convex mirror

     1. A convex mirror can be used as a rear view mirror in automobiles. No doubt the image obtained with a convex mirror is diminished, but the mirror covers a large field of view. Hence, the objects situated behind the automobile and on bothe the sides of road can be easily seen by the driver. 

     2. A convex mirror is also used as a street light reflector. 

Key words

1. Focal length. Distance between pole and the principal focus. 

2. Image (real). A point where reflected rays actually intersect. 

3. Image (virtual). A point where reflected rays appear to intersect. 

4. Magnification (linear). Ratio between size of image to the size of object measured perpendicular to the principal axis. 

5. Magnification (longitudinal). Ratio between size of image to the size of object measured along the direction of principal axis. 

6. Mirror (concave). Spherical reflecting surface buldging at the corners and depressed at the centre. 

7. Mirror (convex). Spherical reflecting surface buldging at the centre and depressed at the corners. 

8. Pole. Centre point of a spherical mirror. 

9. Principal axis. A line passing through the pole of the mirror and perpendicular to its surface. 

10. Principal focus. A point on the principal axis where a beam of light coming parallel to principal axis meets or appears to meet after reflection from the mirror. 

11. Principal section. A section of the mirror by a plane passing through the centre of curvature and the pole. 

12. Radius of curvature. Radius of the sphere of which the mirror forms a part. 

13. Ray. Path along which light travels. 

Key Formulae

1. Relation between focal length (f) and radius of curvature (R) of a mirror :

                        f = R/2

2. Mirror formula : 1/v + 1/u = 1/f

3. Magnification 'm' (linear) : m = I/O = ±v/u

4. Relation between m and f :

                m = f - v/f = f/f - u

5. Magnification (longitudinal) : m' = dv/du

6. Relation between m and m' : m' = - m².