 From M. K. Srivastava of physics department IIT, this is the first lecture of a five lecture series on polarization of light. In these lectures on polarization of light we shall begin with introduction to the Maxwell's equations and electromagnetic waves. We shall then take a problem of polarization of light, what is meant by it, what is unpolarized light, how an unpolarized light beam becomes polarized by refraction or scattering. Then we shall look at Brewster's law, shall try to understand working of a polaroid, then we shall take a plot of Mahler's, we shall consider superposition of two electromagnetic waves under different conditions. We shall consider phenomena of double refraction in crystals and isotopic crystals, try to analyze various situations therein. We shall try to look at the phenomena of dichroism, we shall study the working of a nickel prism, thereafter we shall consider interference of polarized light, working of full wave plate, a half wave plate and quarter wave plate of doubly refracting crystals. We shall consider different types of polarizations and then their analysis. And finally, in the last lecture we shall study the phenomena of optical activity, Fresnel's theory of optical rotation and then the working of polarimeters, number one, Lorentz half-shaped polarimeter and Bichor's polarimeter for the measurement of optical rotation. So, let us begin. You see the light waves are an electromagnetic phenomena. These are electromagnetic waves like x-rays, gamma rays, radio waves etc. You see this figure shows wavelength range of different electromagnetic radiations such as gamma rays as I said, x-rays, ultraviolet radiation, visible light, infrared radiation, short radio waves, radio broadcasting etc. If the wavelength range of the visible light is about 400 to 800 nanometers only as shown in the figure, it covers a very small portion of the large wavelength range of the electromagnetic waves in general. The electric and magnetic fields associated with these waves satisfy Maxwell's equations. These equations enforce certain conditions on the electric field E, the electric displacement D, the magnetic induction B and the magnetic field H and provide connecting relations between them. These equations are number one, Gauss's law, del dx by del x plus del dy by dy plus del dz by dz equal to rho. Here the rho is the charge density in the medium. Number two, absence of monopoles del Bx by del x plus del By by del y plus del Bz by del z equal to 0. Number three, Faraday's law of electromagnetic induction del Ex by del Ey minus del Ey by del Ex is equal to minus del Bz by del t. Similarly for the other three components. Number four, Ampere's law with Maxwell's corrections. This is again a equation relating H, D and J. Here J is the current density in the medium. Using vector notations, if you like that, they have equations can be written in the following compact form, divergence of D is equal to rho, divergence of B is equal to 0 indicating absence of monopoles, curl of E is equal to minus del B by del t that is Faraday's law and curl H is equal to J plus del D by del t that is Ampere's law with Maxwell's corrections. For a linear isotropic and homogeneous medium, this quantity is D, B and J are given by D is equal to sigma time C, B is equal to mu times H, J equal to sigma time C where epsilon, mu and sigma denote respectively the dielectric permittivity, magnetic mayability and conductivity of the medium. For non-charge and current free dielectric that is the source free region where rho is equal to 0 and J is also equal to 0 and the above equations in that case they reduce to divergence of E equal to 0, divergence of H is also equal to 0, curl of E is equal to minus del H by del t and curl of H is equal to epsilon times del E by del t. You see we shall not derive these equations here. Now the above equations you have seen they are coupled equations, they involve both E and H in an equation. The equations satisfied by E or H alone can be obtained by decoupling them by eliminating either H from the equation or E from the equation. The result of this decoupling is del 2 E by del t square is equal to 1 upon mu epsilon times del square E for the electric field and similarly del 2 H by del t square equal to 1 upon mu epsilon del square H for the magnetic field. These are wave equations for the wave travelling with the speed v given by 1 upon square root of mu times epsilon. In free space that is in vacuum no medium the speed is 1 upon square root of mu naught times epsilon naught which is equal to 3 into 10 raised to the power 8 meters per second. It is denoted by C usually the speed of light in vacuum. Now the ratio C by E which is equal to square root of mu epsilon divided by mu naught epsilon naught which is almost equal to the square root of epsilon upon epsilon naught is equal to n the refractive index of the medium. Here mu naught and epsilon naught are respectively the permeability and the electric permittivity of free space and mu is almost equal to mu naught. You see the conditions divergence of E equal to 0 and divergence of H equal to 0 make these waves transverse. This means that E and H do not have longitudinal components. If the wave is propagating along the z axis then E and H have only x and y components now z component they are in the transverse plane. Let us consider the condition curl of E is equal to minus mu times del H by del t and the other one curl H is equal to epsilon times del E by del t. See these conditions they enforce that number 1 E and H are related as H is equal to E divided by a square root of mu by epsilon number 2 they are mutually perpendicular that is if the wave is propagating along the z direction they are perpendicular to the z axis that is they are in the x y plane and number 4 they are in the same phase E and H are in the same phase. The electric field E the magnetic field H and the direction of propagation are this like 3 mutually perpendicular right handed system of axis x y z. Let us now consider the polarization its meaning and what is unpolarized light for plane monochromatic waves traveling along the positive z axis the electric and magnetic fields may be written in the form E is equal to E not cos of kz minus omega t and H is equal to H not cos of kz minus omega t because they are in the same phase here omega is equal to 2 pi by lambda and omega by k is equal to V this velocity and E not and H not are a space and time independent vectors with longitudinal components E not z and H not z equal to 0 as the waves are transverse. In the transverse plane which is the x y plane here the electric field E can have any direction remaining within the x y plane the magnetic field H is anyway perpendicular to it as we have seen and thereby can also have any direction in the x y plane this is the situation in an ordinary light beam of natural light such a beam is called unpolarized the electric vector in an unpolarized beam continues to change its direction although always remaining in the x y plane. So, continues to change its direction in a random manner in intervals of the order of 10 minus 8 seconds every orientation of E is to be regarded as equally probable. So, that as indicated in the figure the average effect is completely symmetrical about the direction of propagation there is still another representation of unpolarized light which is perhaps more useful if it is all the vibration at any instant in the figure into components along the axis A x equal to A cos theta cos of k z minus omega t and A y equal to A sin theta again cos of k z minus omega t. Now these components will in general be unequal but when theta is allowed to assume all values at random the net result is as though we had two vibrations at right angles with equal amplitudes but no coherence of phase essentially each is the result of a large number of individual vibrations with random phase and because of this randomness a complete incoherence is produced. But if the direction of vibration continues to remain unchanged we say that the light is plane polarized or linearly polarized since its vibrations are confined now to the plane containing the z axis which is the direction of application and oriented at some fixed angle theta. This picture shows the electric and magnetic fields of an x poloise light propagating in the z direction. The light shaded region depicts oscillations of the electric field along the x axis but the dark shaded region shows oscillations of the magnetic field along the y axis. It is a poloise beam propagating along the z axis. This is another convenient way of picturing these vibrations. This figure shows a pictorial representation of side views a b and c and n views d e f of plane polarized and unpolarized ordinary light beams. Part a and b represent the two plane poloise components a in the plane of the screen and b perpendicular to the plane of the screen. The part c the two together in an unpolarized beam. Dots represent the end on view of linear vibrations perpendicular to the plane of the screen and double pointed arrows represent vibrations confined to the plane of the screen. It should be noted that the phenomenon that the polarization phenomena it is lack of symmetry in vibrations when viewed against light propagation is a feature of transverse waves only and as we have seen electromagnetic waves or transverse waves. Now we shall consider now polarization by reflection. Consider an unpolarized light beam incident at an angle y on a dialectical like glass I shown in the figure. In general there will always be a reflected ray o r and a reflected ray o t. It is found that these rays reflected and reflected ones are in general partially plane polarized. Partially plane poloise means that in the reflected light there is preponderance of electric vibrations perpendicular to the plane of incidence. the plane of the screen compared to the vibrations parallel to the plane of incidence. On the other hand in the reflected beam there is preponderance of electric vibrations parallel to the plane of incidence. At a certain definite angle of about 57 degrees for ordinary glass the reflected ray is fully plane polarized. It was Brewster who first discovered that at the polarizing angle phi bar the reflected and reflected rays are just 90 degrees apart. This remarkable discovery enables one to correlate polarization with the refractive index n of the medium which is given by sin phi divided by sin phi prime equal to n where phi prime is the angle of reflection. Since at phi bar the angle r o t is 90 degrees we have this sort of relation between phi bar and phi prime bar leading to the condition that the ratio of sin phi bar and cos phi bar becomes equal to n that is n is equal to 10 of phi bar phi bar is the polarizing angle. This is Brewster's law which shows that the angle of incidence for maximum polarization depends only on the refractive index through this relation n is equal to 10 phi bar. This angle is called Brewster's angle or polarizing angle because of its dependence on the refractive index this angle phi bar varies somewhat with the wavelength but for ordinary glass the dispersion is such that the angle phi bar does not change much over the whole of the visible spectrum. At this angle phi bar of incidence if the incident beam is plane polarized with electric vibrations in the plane of incidence there will be no reflected ray the reflected coefficient is 0 in this special situation. It is not difficult to understand the physical reason why the light with vibrations in the plane of incidence is not reflected at Brewster's angle. The incident light sets the electrons in the atoms of the material into oscillations and it is the re-radiation from these electrons that generate the reflected beam. When the later is observed at 90 degrees to the reflected beam only the vibrations that are perpendicular to the plane of incidence can contribute. Those in the plane of incidence have no component transverse to the 90 degree direction which is the direction of the reflected ray and hence cannot radiate in that direction and there is no reflected ray. We shall not take up polarization by refraction. At Brewster's angle we have seen the reflected light is only partially plane polarized with electric vibrations in the plane of incidence dominating over the vibrations perpendicular to the plane of incidence here that is the plane of the screen. One thing should be noted at no angle of incidence whatever the reflected light is fully plane polarized it is always partially plane polarized. If one uses the large number of parallel reflecting surfaces that is a pile of plates the transmitted beam will go on becoming more and more plane polarized and one would obtain an almost plane polarized beam in the end. The figure shows the arrangement of pile of plates the marking is given on the left of the figure with shortening horizontal arrows indicate that the transmitted beam is becoming more and more plane polarized. The degree of polarization p of the transmitted light can be calculated by summing the intensities of the parallel and perpendicular components. If these intensities are called ip and is respectively it can be shown that the degree of polarization p is given by p is equal to ip minus is divided by ip plus is which is equal to m divided by m plus 2 n square on 1 minus n square but m is the number of plates that is 2 m reflecting surfaces and n is their refractive index. This equation shows that by the use of a n of number of plates by making m pretty large the degree of polarization can be made to a probability that is almost 100 percent. Let us now consider polarization by scattering. If an unpolarized beam is allowed to fall on a gas contained in some glass chamber glass container then the beam is scattered at 90 degrees to the incident beam is always found to be plane polarized. This figure shows the unpolarized incident beam and plane polarized scattered beams scattered at right angles to the incident beam. If the incident wave is propagating along the z direction then the scattered wave along the y direction is x polarized and one scattered along the x direction is y polarized. This follows from the fact that the scattered waves propagating in the y direction are produced by the x component of dipole oscillations. The y component of dipole oscillations they produce no field in the y direction you see the y component produces no field in the y direction x component produces no field in the x direction that is the situation. If the incident beam is plane polarized with the selected vector along the x direction then there will be no scattered light along the x axis scattered wave will be there only along the y direction and they will be x polarized. Let us now consider the working of a polarized the usual device used in various experiments to produce plane polarized light or to analyze it is called a polarizer or a polarized. It consists of a sheet of polyvinyl alcohol which has long chain polymer molecules. This sheet is subjected to a large strain which results in orienting these molecules parallel to the strain. These long chain molecules are now almost parallel to each other I mean sort of arrange in a longitudinal fashion. They are then impregnated with iodine which provides high conductivity along the length of the chain along the longitudinal direction. Because of this high conductivity provided by the iodine atoms the electric field parallel to the molecule gets absorbed essentially along that direction the substance will not sustain any electric field. When a light beam is incident on such a polarized the molecules which are aligned parallel to each other we have seen they absorb the component of electric field which is parallel to the direction of alignment because of the high conductivity provided by iodine atoms. While the component having electric field perpendicular to it passes through essentially without any attenuation the device does acts as a polarizer. In actual practice this polymer sheet is then placed between two glass plates so that it is a little protected from scratches etc. This is all in the first lecture.