Plarization

 Polarized light waves are light waves in which the vibrations occur in single plane. The process of transforming unpolarized light into polarized light is known as polarization of light. 

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        From the phenomena of interference and diffraction of light, it is proved that light has a wave nature. However, this does not tell about the type of waves. Light waves are electromagnetic waves in nature that travel through the vacuum. There is a periodic fluctuation in electric and magnetic fields along the propagation of light wave. These fields vary at right angles to the direction of the propagation of the light wave, so light wave is transverse wave. Transverse nature of light make it possible to produce and detect polarized light.

      In transverse mechanical waves, such as produced in a stretched string, the vibrations of the particles of the medium are perpendicular to the direction of propagation of the waves. The direction can be oriented along vertical, horizontal or any other direction as shown in figure given below.

   Fig   Transverse waves on a string polarized (a) in a horizontal plane (b) in a vertical plane.

      In each of these cases, the transverse mechanical wave is said to be polarized. The plane of polarization is the plane containing the direction of the vibration of the particles of the medium and direction of propagation of the wave.

       A light wave produced by oscillating charge consists of a periodic variation of electric field vector accompanied by the magnetic field vector at right angle to each other. Ordinary light has components of vibration in all possible planes. Such a light is unpolarized On the other hand, if the vibrations are confined only in one plane, the light is said to be polarized.

Production and Detection of Plane Polarized Light.

      An ordinary incandescent light emits unpolarized light, as does the sun, because its (electrical) vibrations are randomly oriented in space. It is possible to obtain plane polarized beam of light from unpolarized light by removing all waves from the beam except those having vibrations along one particular direction. This can be achieved by various processes such as selective absorption, reflection from different surfaces, refraction through crystals and scattering by small particles.

      The selective absorption method is the most common method to obtain plane polarized light by using certain types of materials called " dichroic substances ". These materials transmit only those waves, whose vibrations are parallel to a particular direction and will absorb those waves whose vibrations are in other directions. One such commercial polarizing material is a Polaroid.
      If unpolarized light is made incident on a sheet of Polaroid, the transmitted light will be plane polarized. If a second sheet of Polaroid is placed in such a way that the axes of the Polaroid, shown by straight lines drawn on them, are parallel (figure below), the light is transmitted through the second Polaroid. If the second Polaroid is slowly rotated about the beam of the light, as axis of rotation, the light emerging out of the second Polaroid gets dimmer and dimmer and disappears when the axes become mutually perpendicular. The light reappears on further rotation and becomes brightest when the axes are again parallel to each other.

  Fig   Experimental arrangement to show that light waves are transverse.

        This experiment proves that light waves are transverse waves. If the light waves were longitudinal, they would never disappear even if the two Polaroids were mutually perpendicular.
        Reflection of light from water, glass, snow and rough road surfaces, for larger angles of incidences, produces glare. Since the reflected light is partially polarized, glare can considerably be reduced by using Polaroid sunglasses.
      Sunlight also becomes partially polarized because of scattering by air molecules of the Earth's atmosphere. This effect can be observed by looking directly up through a pair sunglasses made of polarizing glass. At certain orientations of the lenses, less light passes through than at others.

Optical Rotation

      When a plane polarized light is passed through certain crystals, they rotate the plane of polarization. Quartz and sodium chlorate crystals are typical examples, which are termed as optically active crystals.
     A few millimeter thickness of such crystals will rotate the plane of polarization by many degrees. Certain organic substances, such as sugar and tartaric acid, show optical rotation when they are in solution can be used to determined their concentration in the solutions.

  Fig   Solution of an optical isomer rotates the plane of polarization of incident light so that it is no  longer horizontal but at an angle. The analyzer thus stops the light when rotated from the vertical (cross) positions.

Video for Polarization of Light

Diffraction of X-Rays by Crystals

X-rays are diffracted by crystals in a manner dependent on the wavelength of the rays and the space lattice of the crystal. Thus X-ray diffraction provides a means for study of the structure of crystalline substances, or of substances which have crystalline phases. The method adopted depends on the form in which the substance is available. With large crystals Laue diagrams can provide useful characterization, but more frequently the crystal is rotated when mounted at the center of a cylindrical film, thus bringing successive sets of crystalline planes into position. The Debye-Scherrer ring or powder method is used when the specimen consists of a number of small crystals. Because of the number of crystals, randomly distributed, some are usually available in each plane to the diffract the X-ray beam.

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        X-rays is a type of electromagnetic radiation of much shorter wavelength, about m. In order to observe the effects of diffraction, the grating spacing must be of the order of the wavelength of the radiation used. The regular array of the atoms in a crystal forms a natural diffraction grating with spacing that is typically m. The scattering of X-rays from the atoms in a crystalline lattice gives rise to diffraction effects very similar to those observed with visible light incident on ordinary grating.

       The study of atomic structure of crystals by X-rays was initiated in 1914 by W. H. Bragg and W. L. Bragg with remarkable achievements. They found that a monochromatic beam of X-rays was reflected from a crystal plane as if it acted like mirror. To understand this effect, a series of atomic planes of constant inter planer spacing d parallel to a crystal face are shown by lines PP', P1P1', P2P2' and so on, in the figure given below:

   Fig    Diffraction of X-rays from the lattice plane of crystal

      Suppose an X-rays beam is incident at an angle θ on one of the planes. The beam can be reflected from both the upper and the lower planes of atoms. The beam reflected from lower plane travels some extra distance as compared to the beam reflected from the upper plane. The effective  path difference between the two reflected beams is 2dsinθ. Therefore, for reinforcement, the path difference should be an integral multiple of the wavelength. Thus

2dsinθ =nλ

      
       The value of n is referred to as the order of reflection. The above equation is known as the Bragg equation. It can be used to determine inter planar spacing between similar parallel planes of a crystal if X-rays of known wavelength are allowed to diffract from the crystal.

    X-ray diffraction has been very useful in determining the structure of biologically important molecules such as hemoglobin, which is an important constituent of blood, and double helix structure of DNA. 

Diffraction Grating

 A diffraction grating is a glass plate having a large number of close parallel equidistant slits mechanically rules on it. The transparent spacing between the scratches on the glass plate act as slits. A typical diffraction grating has about 400 to 5000 lines per centimeter. 

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        In order to understand how a grating diffracts light, consider a parallel beam of monochromatic light illuminating the grating at normal incidence. A few of the equally spaced narrow slits are shown in the figure given below.

   Fig   Diffraction of light due to grating

        The distance between two adjacent slits is d, called grating element. Its value is obtained by dividing the length L of the grating by the total number N of the lines rules on it. The sections of wavefront that pass through the slits behave as source of secondary wavelets according to Huygen's principle.

     In the above figure, consider the parallel rays which after diffraction through the grating make an angle θ with AB, the normal to the grating. They are then brought to focus on the screen at P by a convex lens. If the path difference between ray 1 and 2 is one wavelength λ, they will reinforce each other at P. As the incident beam consists of parallel rays, the rays from any two consecutive slits will differ in path by λ when arrive at P. They will, therefore, interfere constuctively. Hence, the condition for constructive interference is that ab, the path difference between two consecutive rays, should be equal to λ, i.e.,

ab = λ 

From figure,

ab = dsinθ 

d being the grating element. Substituating the value of ab in the above equation we get,

dsinθ = λ 

        According to the above equation, when  θ = 0 i.e., along the direction of normal to the grating, the path difference between the rays coming out from the slits of the grating will be zero. So we will get a bright image in this direction. This is known as zero-order image formed by the grating.

        If we increase θ on either side of this direction, a value of θ will be arrived at which dsinθ will become λ and according to the dsinθ = λ, we will again get a bright image. This is known as the first-order image of the grating. In this way if we continue increasing θ, we will get the second, third, etc. images on either side of the zero-order image with dark regions in between. The second, third order bright images would occur according as dsinθ becoming equal to 2λ, 3λ, etc. Thus the equation dsinθ = λ can be written in more general form as,

dsinθ =nλ 

where n = 0,1,2,3, etc.

        However, if the incident light contains different wavelengths, the image of each wavelength for certain value of n is diffracted in a different directions. Thus, separate images are obtained corresponding to each wavelength or color. The above equation shows that the value of θ depends upon n, so the images of different colors are much separated in higher orders. 

Diffraction due to a narrow slit

As light bends around an obstacle this phenomena can also be observe by passing light through a narrow slit which proves that light ray deviates from its straight path while passing from sharp edge. 

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        The figure below, shows the experimental arrangement for studying the diffraction of light due to a narrow slit.

    Fig    Diffraction due to a narrow slit AB

      The slit AB width d is illuminated by a parallel beam of monochromatic light of wavelength λ. The screen s is placed parallel to the slit for observing the effects of the diffraction of light. A small portion of the incident wavefront passes through the narrow slit. Each point of this section of the wavefront sends out secondary wavelets to the screen. These wavelets then interfere to produce the diffraction pattern. It becomes simple to deal with rays instead of wavefronts as shown in the figure. In the above figure, only seven rays have been drawn whereas actually there are a large number of them. Let us consider rays 1 and 4 which are in phase when in the wavefront AB. After these reach the wavefront AC, ray 4 would have path difference ab say equal to λ/2. Thus, when these two rays reach point P on the screen they will interfere destructively. Similarly, each pair 2 and 6, 3 and 7 differ in path by λ/2 and will do the same. But the path difference ab = d/2 sinθ.

       The equation for the first minimum is, then

      In general, the condition for different orders of minima (dark regions) on either side of center are given by,

                 where m = 1,2,3,...

      
       The region between any two consecutive minima both above and below O will be bright. A narrow slit, therefore, produces a series of bright and dark regions with the first bright region at the center of the pattern. Such a diffraction pattern is shown in figures given below:

         Diffraction due to monochromatic light                       Diffraction due to white light        t

Diffraction of Light

The property of bending of light around obstacles and spreading of light waves into the geometrical shadow of an obstacle is called diffraction. 
If the shadow of an object cast on a screen by a small source of light is examined, it is found that the boundary of the shadow is not sharp. The light is not propagated strictly in straight lines. This phenomena of diffraction, which occurs as the light passes the object, results from the wave nature of light. Banded or annular patterns, diffraction patterns, are produced near the edges of the shadow. Small apertures in objects produce a similar effect.  

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      In Young's double slit experiment for the interference of light, the central region of the fringe system is light. If light travels in a straight line, the central region should appear dark i.e. the shadow of the screen between the two slits. Another simple experiment can be performed for exhibiting the same effect.
       Consider that a small and smooth steel ball of about 3mm in diameter is illuminated by a point source of light. The shadow of the object is received on a screen as shown in the figure given below. The figure shows that the shadow of the spherical object is not completely dark but has a bright spot at its center. According to Huygen's principle, each point on the rim of the sphere behaves as a source of secondary wavelets which illuminate the central region of the shadow.

Bending of light caused by its passage past a spherical object

    These two experiments clearly show that when light travels past an obstacle, it does not proceed exactly along straight path, but bends around the obstacle.
       The phenomena is found to be prominent when the wavelength of light is large as compared with the size of the obstacle or aperture of the slit. The diffraction of light occurs, in effect, due to the interference between rays coming from different parts of the same wavefront.

                    Their colors are due to diffraction                   .

Michelson's Interferometer

In order to verify the wave nature of light and to measure the wavelength of light or to observe interference of light, Michelson invented his famous interferometer in which he divide the light from single source into two parts with the help of semi silvered glass plate.  

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      The Michelson interferometer was invented by an American physicist A.A.Michelson (1852-1931). The Michelson interferometer played an interesting role in the history of science during the latter part of the nineteenth century. It has a great scientific importance and had an equally important role in establishing high precision standards of the unit of length. In contrast to the Young's double slit experiment for producing interference fringes which make use of light from two narrow sources, the Michelson interferometer uses light from broad, spread source (extended source).
    The essential features of a Michelson interferometer are shown schematically in figure given below:

   Fig    Schematic diagram of Michelson Interferometer

     Michelson interferometer consists of two highly polished plane mirrors M1 and M2. The mirror M1 is fixed where as the mirror M2 is moveable as shown in the above figure. In addition to this, it has glass plate C which has a thin coating of silver on its right side. This partially silvered plate is called beam splitter and is inclined at 45° relative to the incident light beam. It has also another plate D which is identical to the plate C except it is not silvered. Its purpose is to ensure that the beam I and II pass through the same thickness of glass. Therefore it is known as compensating plate. This is particularly important when white light fringes (colored fringes) are desired.

Newton's Rings

Circular interference formed between a lens and a glass plate with which the lens is in contact. There is a central dark spot around which there are concentric dark fringes.The radius of the nth ring is given by . Where λ is the wavelength and R is the radius of curvature of the lens. 

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      When a Plano convex lens of long focal length is placed in contact on a plane glass plate (Figure given below), a thin air film is enclosed between the upper surface of the glass plate and the lower surface of the lens. The thickness of the air film is almost zero at the point of contact O and gradually increases as one proceeds towards the periphery of the lens. Thus points where the thickness of air film is constant, will lie on a circle with O as center.

        By means of a sheet of glass G, a parallel beam of monochromatic light is reflected towards the lens L. Consider a ray of monochromatic light that strikes the upper surface of the air film nearly along normal. The ray is partly reflected and partly refracted as shown in the figure. The ray refracted in the air film is also reflected partly at the lower surface of the film. The two reflected rays, i.e. produced at the upper and lower surface of the film, are coherent and interfere constructively or destructively. When the light reflected upwards is observed through microscope M which is focused on the glass plate, series of dark and bright rings are seen with center as O. These concentric rings are known as " Newton's Rings ".
      At the point of contact of the lens and the glass plate, the thickness of the film is effectively zero but due to reflection at the lower surface of air film from denser medium, an additional path of λ/2 is introduced. Consequently, the center of Newton rings is dark due to destructive interference.

   Fig  . Experimental arrangement for observing Newton's rings

      Let us consider a system of plano-convex lens of radius of curvature R placed on flat glass plate it is exposed to monochromatic light of wavelength λ normally.

Interference in thin films

Thin films (e.g. soap bubbles,oil on water) often display brilliant coloration when reflecting white light and show fringes when in monochromatic light.
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      A thin film is a transparent medium whose thickness is comparable with the wavelength of light. Brilliant and beautiful colors in soap bubbles and oil film on the surface of water are due to interference of light reflected from the two surfaces of the film as explained below:

       Consider a thin film of a reflecting medium. A beam AB of monochromatic light of wavelength λ is is incident on its upper surface. It is partly reflected along BC and partly refracted into the medium along BD. At D it is again partly reflected inside the medium along DE and then at E refracted along EF as shown in the figure given below:

   Fig  . Geometrical construction of interference of light due to a thin oil film

        The beams BC and EF, being the parts of the same beam have a phase coherence. As the film is thin, so the separation between the beam BC and EF will be very small, and they will superpose and the result of their interference will be detected by the eye. It can be seen in the above figure, that the original beam splits into two parts at the point B and they inter the eye after covering different lengths of paths. Their path difference depends upon (i) thickness and nature of the film (ii) angle of incidence. If the two reflected waves reinforce each other, then the film as seen with help of a parallel beam of monochromatic light will look bright however, if the thickness of the film and angle of incidence are such that the two reflected waves cancel each other, the film will look dark.
        If white light is incident on a film of irregular thickness at all possible angles, we should consider the interference pattern due to each spectral color separately. It is quite possible that at a certain place on the film, its thickness and the angle of incidence of light are such that the condition of destructive interference of one color is being satisfied. Hence, that portion of the film will exhibit the remaining constituent colors of the white light as shown in the above figure.

        From the above figure incident light first reflects from upper surface gives Part I (pink ray) and also refracts  into the film which again reflects from the bottom surface and comes to the eye as Part II (violet ray).
         Part I of light has phase change of 180° as it is reflected from a surface beyond which there is medium of higher refractive index. But Part II of light has no phase change as it is reflected from a surface beyond which there is a medium of lower index. Therefore the condition for constructive and destructive interference are reversed then the Young's double slit experiment. For nearly normal incidence the path difference between the two interfering rays is twice the thickness of the film i.e equal to 2t where t is the thickness of the film. If n is the refractive index of medium of the film then,

Path difference = 2tn

Hence condition for the maxima or constructive interference is,

                  m = 0,1,2,....

similarly condition for the minima or destructive interference is,

                           m = 0,1,2,....

     In case of varying thickness of film, there will be a pattern of alternate dark and bright fringes. 

   Fig  . Interference pattern produced by thin soap bubbles

Young's Double Slit Experiment - Division of the wavefronts

Thomas Young was the first who observed the interference of light in 1801. He placed monochromatic source of light in front of a single slit. In order to get a coherent light he placed another two slits very close to each other in front of the first one. After passing light rays from the two slits a pattern of alternate dark and bright fringes is formed on a screen placed parallel to slits at some distance.The central fringe was bright. From the interference of light we can easily calculate the wavelength of the light.
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  Simple ray geometry of Young's double slit experiment  

      The above figure shows the experimental arrangement, similar to that devised by Young in 1801, for studying interference effect of light. A screen having two narrow slits is illuminated by a beam of monochromatic light. The portion of the wavefront incident on the slits behaves as a source of secondary wavelets (Huygen's principle). The secondary wavelets leaving the slits are coherent. Superposition of these wavelets result in a series of alternate bright and dark fringes which are observed on a second screen placed at some distance parallel to the first screen.

Interference Of Light Waves

"Superposition of two light waves having phase coherence traveling in the same direction results in a phenomena called Interference."
Light from a Source S passes through a pinhole and falls on two further pinholes A and B, light and dark fringes appear on the screen where the resultant pencils of light overlap. The pencils of light interfere and produce "Interference fringes", first observed by Thomas Young in 1801. These and similar fringes were used by Fresnel and Young to establish the wave theory of light.
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  Fig   Young's Double slit Experiment for Interference

        When two waves are allowed to superpose upon each other and, if the resultant intensity of the interfering waves is zero or less than the intensity of the either individual wave then this type of interference is called "Destructive Interference" and it occurs where crest of one wave falls upon trough of other wave.

       Similarly, if the resultant intensity of the interfering waves is greater than the intensity of an individual wave then this type of interference is known as "Constructive Interference" and this occurs where crest of one wave overlaps the crest of other wave or trough of one wave overlaps the trough of other wave.

       

Essential Conditions for the Interference:

         Interference of light waves is not easy to observe because of the random emission of light from a source. The following conditions must be met, in order to observe the phenomena:

1- The interfering beams must be monochromatic that is, of a single wavelength.

2- The interfering beams must be coherent.

Coherent: waves that are in phase both temporally and spatially. Most practical radiation sources are not coherent over an appreciable length of time since waves trains of limited length are emitted at random intervals. The laser is a source of coherent radiations.

      Consider two or more sources of light waves of the same wavelength. If the sources send out crests or troughs at the same instant, the individual waves maintain a constant phase difference with one another. The monochromatic sources of light which emit waves, having a constant phase difference are called coherent source.

     A common method of producing two coherent light beams is to use a monochromatic source to illuminate a screen containing two small holes, usually in the shape of slits. The light emerging from the two slits is coherent because a single source produces the original beam and two slits serve only to split it into two parts. The points on Huygen's wavefront which send out secondary wavelets are also coherent sources of light.

"In an Interferometer, fringes are produced and used to make accurate measurement of wavelength."

Huygens's Principle

A wave theory of light based on the concept of wavelets or secondary waves spreading from each point affected by a disturbance and conspiring to give a fresh wavefront which envelops the wavelets which create it. The amplitude in a secondary wavelet falls off in proportion to ( 1 + Cosθ ), where θ is the angle with the forward direction. Huygens conceived the waves as longitudinal in nature and could not explain polarization.
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         Knowing the shape and location of a wavefront at any instant t, Huygen's principle enables us to determine the shape and location of the new wavefront at a later time t + ∆t. This principle consists of two parts: 

 Every point of a wavefront may be considered as a source of secondary wavelets which spread out in forward direction with a speed equal to the speed of propagation of the wave.

The new position of the wavefront after a certain interval of time can be found by constructing a surface that touches all the secondary wavelets.

     The principle is illustrated in Fig 1 given below. AB represent the wavefront at any instant t. To determine the wavefront at time t + ∆t, draw secondary wavelets with center at various points on the wavefront AB and radius as c∆t where c is the speed of the propagation of the waves as shown in Fig 1. The new wavefront at time t + ∆t is A'B' which is a tangent envelope to all the secondary wavelets. Fig 2 shows a similar construction for a plane wavefront.

  Fig 1 . Spherical Wavefront                      Fig 2 . Plane Wavefront 

Wavefronts

The surface over which particles are vibrating in the same phase. The surface is normal to rays in isotropic media.
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Isotropic: A body is isotropic if its properties are the same in all direction

       Consider a point source of light as S ( Figure given below ). Waves emitted from this source will propagate outwards in all directions with speed c (c is the speed of light). After time t, they will reach the surface of a sphere with center as S and radius ct. Every point on the surface of this sphere will be set into vibration by the waves reaching there. As the distance of all these points from the source is the same, so their state of vibration will be identical. In other words we can say that all the points on the surface of the sphere will have the same phase.

Phase: Particles in periodic motion due to the passage of a wave are said to be in the same phase of vibration if they are moving in the same direction with the same relative displacement. Particles in a wavefront are in the same phase of vibration and the distance between the phases are the same is the wavelength i.e λ.

   Fig  . Spherical wavefronts

Such a surface on which all the points have the same phase of vibration is known as wavefronts.

         Thus in case of a point source, the wavefront is spherical in shape. A line normal to the wavefront including the direction of motion is called a ray of light.

          With time, the wave moves farther giving rise to new wave fronts. All these wavefronts will be concentric spheres of increasing radii as shown in the figure given above. Thus the wave propagates in space by the motion of the wavefronts is one wavelenth. It can be seen that as we move away at greater distance from the source, the wavefronts are parts of spheres of very large radii. A limited region taken on such a wavefront can be regarded as a plane wavefront ( Shown in figure given below ). For example, light from the sun reaches the Earth in plane wavefronts.

   Fig  . Plane wavefronts

      
       In the study of interference and diffraction, plane waves and plane wavefronts are considered. A usual way to obtain a plane wave is to place point source of light at the focus of a convex lens. The rays coming out of the lens will constitute plane waves.