RAY OPTICS AND OPTICAL INSTRUMENTS

                   1. Introduction

This topic mainly deals with the ray optics (study of light) and the the devices to measure it. This study about light is known as quantum mechanics. Neil Bohr have once said that the person who says that he understand or knows quantum mechanics then the person is either lying or me is making you fool. So, the study of quantum mechanics is not that much simple. Nature has endowed the human eye (retina) with the sensitivity to detect electromagnetic waves within a small range of the electromagnetic 

spectrum. Electromagnetic radiation belonging to this region of the spectrum (wavelength of about 400 nm to 750 nm) is called light. It is mainly through light and the sense of vision that we know and interpret the world around us. 

There are two things that we can intuitively mention about light from common experience. First, that it travels with enormous speed and second, that it travels in a straight line. It took some time for people to realise that the speed of light is finite and measurable. Its presently accepted value in vacuum is c = 2.99792458 × 10^8  m/s. For many purposes, it suffices to take

 c = 3 × 10^8 m/s . The speed of light in vacuum is the highest speed attainable in nature. The intuitive notion that light travels in a straight line seems to contradict, that light is an 

electromagnetic wave of wavelength belonging to the visible part of the spectrum. How to reconcile the two facts? The answer is that the 

wavelength of light is very small compared to the size of ordinary objects that we encounter commonly (generally of the order of a few cm or larger). In this situation, the light wave can be 

considered to travel from one point to another, along a straight line joining them. The path is called a ray of light, and a bundle of such rays

constitutes a beam of light. The phenomena of reflection, refraction and dispersion of light, using the ray picture of light. Using the basic 

laws of reflection and refraction, we shall study the image formation by plane and spherical reflecting and refracting surfaces. We then go on to describe the construction and working of some important optical instruments, including the human eye. 

2. REFLECTION OF LIGHT BY SPHERICAL MIRRORS

We are familiar with the laws of reflection. The angle of reflection (i.e., the angle between reflected ray and the normal to the reflecting surface or the mirror) equals    

    

 the angle of incidence (angle between incident ray and the normal). Also that the incident ray, reflected ray and the normal to the reflecting surface at the point of incidence lie in the same plane give in the above figure. These laws are valid at each point on any reflecting surface whether plane or curved. However, we shall restrict our discussion to the special case of curved surfaces, that is, spherical surfaces. The normal in  this case is to be taken as normal to the tangent to surface at the point of incidence. That is, the normal is along the radius, the line joining the centre of curvature of the mirror to the point of incidence. We have already studied that the geometric centre of a spherical mirror is called its pole while that of a spherical lens is called its optical centre. The line joining the pole and the centre of curvature of the spherical mirror is known as the principal axis. In the case of spherical lenses, the principal axis is the line joining the optical centre with its principal focus as you will see later. 

             3. Sign convention

Sign convention we can easily understand as we represent different points in the cartesian plane as the right side of the origin is the positive side while the left side of the origin is the negative side and similarly the upward side of the origin is the positive side and the downward side is the negative side. And similarly to this we can compare the sign convention which we are going To study now. To derive the relevant formulae for reflection by spherical mirrors and refraction by spherical lenses, we must first adopt a sign convention for measuring distances. In this book, we shall follow the Cartesian sign convention. According to this convention, all distances are measured from the pole of the mirror or the optical centre of the lens. The distances measured in the same direction as the incident light are taken as positive and those measured in the direction opposite to the direction of incident

light are taken as negative. The heights measured upwards with respect to x-axis and normal to the principal axis (x-axis) of the mirror/lens are taken as positive. The heights measure downwards are taken as negative. With a common accepted convention, it turns out that a single formula for spherical mirrors and a single formula for spherical lenses can handle all different cases.

4. Focal length of spherical mirrors

When a parallel beam of light is incident on (a) a concave mirror, and (b) a convex mirror. We assume that the rays are paraxial, i.e., they are incident at points close to the pole P of the mirror

and make small angles with the principal axis. The reflected rays converge at a point F on the principal axis of a concave mirror. For a convex mirror, the reflected rays appear to diverge from a point F on its principal axis [Fig. 9.3(b)]. The point F is called the principal focus of the mirror. If the parallel paraxial beam of light were incident, making some angle with the principal axis, the reflected rays would converge (or

appear to diverge) from a point in a plane through F normal to the principal axis. This is called the focal plane of the mirror.  The distance between the focus F and the pole P of the mirror is called the focal length of the mirror, denoted by f. We now show that f = R/2, where R is the radius of curvature of the mirror. The geometry 

of reflection of an incident ray. Let C be the centre of curvature of the mirror. Consider a 

ray parallel to the principal axis striking the mirror at M. Then CM will be perpendicular to the mirror at M. Let θ be the angle of incidence, and MD be the perpendicular from M on the 

principal axis. Then,

          ∠MCP = θ and ∠M F P = 2θ

Now, tan θ = MD/CD and tan 2θ = MD/FD 

For small θ, which is true for paraxial rays, tan θ ≈ θ, tan 2θ ≈ 2θ. Therefore, MD/FD = 2 MD/CD or, FD = CD/2  

Now, for small θ, the point D is very close to the point P. Therefore, FD = f and CD = R. Then gives 

f = R/2 

           5. The mirror equation

If rays emanating from a point actually meet at another point after reflection and/or refraction, that point is called the image of the first point. The image is real if the rays actually converge to the point; it is virtual if the rays do not actually meet but appear to diverge from the point when produced backwards. An image is thus a point-to-point correspondence with the object established through reflection and/or refraction. In principle, we can take any two rays emanating from a point on an object, trace their paths, find their point of intersection and thus, obtain the image of the point due to reflection at a spherical mirror. In practice, however, it is convenient to choose any two of the following rays: 

(i) The ray from the point which is parallel to the

principal axis. The reflected ray goes through

the focus of the mirror.

(i i) The ray passing through the centre of

curvature of a concave mirror or appearing to pass through it for a convex mirror. The reflected ray simply retraces the path. 

(i i i) The ray passing through (or directed towards) the focus of the concave mirror or appearing to pass through (or directed towards) the focus of a convex mirror. The reflected ray is parallel to the principal axis. 

(i v) The ray incident at any angle at the pole. The reflected ray follows laws of reflection. 

The ray diagram considering three rays. It shows

the image A′B′ (in this case, real) of an object AB formed by a concave mirror. It does not mean that only three rays emanate from the point A. 

An infinite number of rays emanate from any source, in all directions. Thus, point A′ is image point of A if every ray originating at point A and 

falling on the concave mirror after reflection passes through the point A′. We now derive the mirror equation or the relation between the object distance (u), image distance (v) and the focal length (f ). From the two right-angled triangles A′B′F and M P F are similar. (For paraxial rays, MP can be considered to be a straight line perpendicular to CP.) Therefore, 

                      B 'A '/PM = B ' F/FP 

or

                 B 'A '/BA = B 'F /FP     ...........     (PM = AB) 

Since ∠ APB = ∠ A′P B′, the right angled triangles A′B′P and A B P are also similar. 

                     5. REFRACTION

When a beam of light encounters another transparent medium, a part of light gets reflected back into the first medium while the rest enters the other. A ray of light represents a beam. The direction of propagation of an obliquely incident (0°< i < 90°) ray of light that enters the other medium, changes at the interface of the two media. This phenomenon is called refraction of light. Snell experimentally obtained the following laws of refraction: 

(i) The incident ray, the refracted ray and the

normal to the interface at the point of

incidence, all lie in the same plane.

(i i) The ratio of the sine of the angle of incidence

to the sine of angle of refraction is constant.

Remember that the angles of incidence (i ) and

refraction (r ) are the angles that the incident

and its refracted ray make with the normal, respectively. We have

sin i / sin r = n 21

where n 21 is a constant, called the refractive index of the second medium with respect to the first medium.The well-known Snell’s law of refraction. We note that n 21 is a characteristic of the pair of media (and also depends on the wavelength of light), but is independent of the 

angle of incidence. If n 21 > 1, r < i, i.e., the refracted ray bends towards the normal. In such a case medium 2 is said to be optically denser (or 

denser, in short) than medium 1. On the other hand, if n 21 <1, r > i, the refracted ray bends away from the normal. This is the case when incident ray in a denser medium refracts into a rarer medium. The refraction of light through the atmosphere is responsible for many interesting phenomena. For example, the Sun is visible a little before the actual sunrise and until a little after the actual sunset due to refraction of light through the atmosphere. By actual sunrise we mean the actual crossing of the horizon by the sun. The actual and apparent positions of the Sun with respect to the horizon. The figure is highly 

exaggerated to show the effect. The refractive index of air with respect to vacuum is 1.00029. Due to this, the apparent shift in the direction of the Sun is by about half a degree and the corresponding time difference between actual sunset and apparent sunset is about 2 minutes. The apparent flattening (oval shape) of the Sun at sunset and sunrise is also due to the same phenomenon.