UNIT 2

OPTICS

2.1 INTERFERENCE

Interference in light waves occurs whenever two or more waves overlap at a given point.

TYPES OF INTERFERENCE

The interference of light waves can be either constructive interference or destructive interference.

• Constructive Interference: Constructive interference takes place when the crest of one wave falls on the crest of another wave such that the amplitude is maximum. These waves will have the same displacement and are in the same phase.
• Destructive Interference: In destructive interference, the crest of one wave falls on the trough of another wave such that the amplitude is minimum. The displacement and phase of these waves are not the same.

Figure 1: Constructive interference & destructive interference

We know that the superposition of two mechanical waves can be constructive or destructive. In constructive interference, the amplitude of the resultant wave at a given position or time is greater than that of either individual wave, whereas, in destructive interference, the resultant amplitude is less than that of either individual wave.

Light waves also interfere with each other. Fundamentally, all interference associated with light waves arises when the electromagnetic fields that constitute the individual waves combine.

If two light bulbs are placed side by side, no interference effects are observed because the light waves from one bulb are emitted independently of those from the other bulb. The emissions from the two light bulbs do not maintain a constant phase relationship with each other over time. Light waves from an ordinary source such as a light bulb undergo random phase changes in time intervals less than a nanosecond.

Therefore, the conditions for constructive interference, destructive interference, or some intermediate state are maintained only for such short time intervals. Because the eye cannot follow such rapid changes, no interference effects are observed. Such light sources are said to be incoherent.

COHERENT SOURCES

Two sources are said to be coherent when the waves emitted from them have the same frequency and constant phase difference.

Interference from such waves happen all the time, the randomly phased light waves constantly produce bright and dark fringes at every point. But, we cannot see them since they occur randomly. A point that has a dark fringe at one moment may have a bright fringe at the next moment. This cancels out the effect of the interference effect, and we see only an average brightness value. The interference is not said to be sustained since we cannot observe it.

Definition:- A predictable correlation of the amplitude and phase at any one point with another point is called coherence.

Figure 2: Incoherence & coherence

Two waves are said to be coherent, the waves must have

 In the case of conventional light, the property of coherence exhibits between a source and its virtual source whereas in the case of laser the property coherence exists between any two or more light waves.

There are two types of coherence

i)                   Temporal coherence

ii)                Spatial coherence

TEMPORAL COHERENCE (OR LONGITUDINAL COHERENCE):-

SPATIAL COHERENCE (OR TRANSVERSE COHERENCE)

The predictable correlation of amplitude and phase at one point on the wave train w. r .t another point on a second wave, then the waves are said to be spatial coherence (or transverse coherence)

Figure 4: Spatial Coherence

CONDITIONS OF INTERFERENCE

In a sustained interference pattern, the position of maximum and minimum intensity regions remains constant with time. To obtain sustained interference, the following conditions are required:

• The sources must be coherent, that is, they must maintain a constant phase with respect to each other.
• The two light sources must emit continuous waves of the same wavelength and have the same period.
• The distance between the two sources of light must be small. This gives large fringe width so that the fringes are separately visible.
• The sources should be monochromatic, that is, of a single wavelength.
• The two light sources must emit waves in nearly the same direction.
• The light source must be a point source.
• The distance between the two sources and the screen must be large. This gives again large fringe width so that the fringes are separately visible.

Key Takeaways

1. Interference in light waves occurs whenever two or more waves overlap at a given point.
2. The interference of light waves can be either constructive interference or destructive interference.
3. To obtain sustained interference; the sources must be coherent.
4. There are two types of coherence: Temporal coherence & Spatial coherence

2.2 NEWTON’S RINGS

Newton’s Rings are the circular interference pattern first discovered by physicist Sir Isaac Newton in 1704. It is consists of concentric bright and dark rings with the point of contact of the lens and the glass plate as centre,

The fringes obtained by interference of light waves by using the following arrangement

When a Plano-convex lens with a large radius of curvature is placed on a plane glass plate such that its curved surface faces the glass plate, a wedge air film (of gradually increasing thickness) is formed between the lens and the glass plate. The thickness of the air film is zero at the point of contact and gradually increases away from the point of contact.

Figure 5: Newton Ring Assembly

If monochromatic (means light with single wavelength) light is allowed to fall normally on the lens from a source 'S', then two reflected rays R1 (reflected from the upper surface of the film) and R2 (reflected from the lower surface of the air film) interfere to produce a circular interference pattern. This interference pattern has concentric alternate bright and dark rings around the point of contact. This pattern is observed through a traveling microscope.

Key Takeaways

1. Newton’s Rings consists of concentric bright and dark rings with the point of contact of the lens and the glass plate as centre.
2. The diameter of the bright ring is proportional to the square root of odd natural numbers D=  n=0,1,2,3,4
3. The diameter of the dark ring is proportional to the square root of natural numbers Dn=               n=0,1,2,3,4
4. The Newton’s rings are not equally spaced because the diameter of the ring does not increase in the same proportion as the order of ring and rings get closer and closer as ‘n’ increases.

2.3 ENGINEERING APPLICATIONS OF INTERFERENCE I) DETERMINATION OF REFRACTIVE INDEX OF LIQUID  2) TESTING OF OPTICAL FLATNESS

Key Takeaways

2.4 MICHELSON'S INTERFEROMETER AND IT'S APPLICATION FOR DETERMINATION OF REFRACTIVE INDEX OF THIN FILM

INTERFEROMETERS

Interferometry is a word used to study and exploit interference of light in optical devices in which two or more beams originate from the same source and after passing through different paths, they interfere with each other. The interference pattern can be analyzed to get information about the objects or phenomenon being studied. This idea involves the principle of division of amplitude.

An interferometer is designed to make the use of two-beam interference (Michelson Interferometer) and multiple-beam interference (Fabry-Perot Interferometer) of light.

There are two types of interferometers:

(1) Wavefront division interferometers in which the different portions of the same wavefront of a coherent beam of light is used to produce interference (Young's double slit, Lloyd's mirror, Fresnel Biprism, etc.) and

(2) Amplitude division interferometers in which a beam splitter is used to divide the initial coherent beam into two parts.

Basic Principle of Interferometers

When a beam of light is incident on a beam splitter or half-transparent mirror (a glass plate partially coated with sliver acts as a partially reflective surface), the incident light beam splits into two beams; one is directed towards a fixed mirror and the other is transmitted towards another movable mirror. The two beams reflected from these mirrors are subsequently recombined and produce interference pattern due to phase or path difference between them. An optical device that works on the above said principle is called an interferometer.

It is a scientific tool working on the principle of interference of light waves for making precise measurements with extraordinary accuracy such as the surface irregularities of the optical elements like a lens, mirrors, and so on, to measure the thickness of a thin wafer in terms of the wavelength of light, to study spectrum lines and for the determination of refractive indices of transparent materials as well as gases. In astronomy, interferometers are used to measure the distances between stars and the diameters of stars.

The Michelson Interferometer

The Michelson interferometer, first designed by Albert Michelson in 1881, played a vital role in the development of modem physics. The interferometer was used to establish experimental evidence for the validity of the special theory of relativity, to detect and measure hyperfine structure in line spectra, to provide a substitute standardization of a meter in terms of wavelengths of light, and to measure the wavelengths of the two main lines of the sodium spectrum and the refractive index of air, etc.

Figure 9: Michelson Interferometer Experiment

Principle

When a beam of light is incident on a beam splitter, it splits up into two beams. One is directed towards a fixed mirror and the other is transmitted towards another movable mirror. The two beams are reflected from the mirrors and are subsequently recombined to produce an interference pattern due to phase or path difference between them.

Construction:

A schematic diagram of the Michelson interferometer is shown in Figure 10. It consists of an extended light source S, a partially coated glass plate G1 called a beam splitter, two highly polished plane mirrors M1and M2, a compensating glass plate G2, and a telescope T. The glass plate G1 is placed at an angle of 45° to the directions of AM1 and AM2 which are perpendicular to each other. The mirror is mounted on a fine pitch screw having a least count of about 10-5 cm. The mirror M1 can be moved parallel to itself by this vernier arrangement. The mirrors M1 and M2 are provided with level fixing screws at their back and with the help of these screws, the mirrors can be made exactly perpendicular to each other or any other desired position.

Figure 10: Michelson interferometer

Working

A light beam from the source S falls on the beam splitter G1. It splits into two perpendicular beams AM1 and AM2 of equal intensity. The light beam SA is partially reflected at the back of the glass plate G1 to produce the beam AM1 and it is partially refracted through G1 to produce the beam AM2. The two beams are then reflected along their original path by two separate mirrors M1 and M2 which are located at different distances from the beam splitter. The reflected beam M1 will be transmitted through G1 while the reflected beam M1A suffers reflection at the back of G1 and both reach the telescope T.

Thus, the telescope receives two reflected beams which are originally derived from the same source so that these are mutually coherent. Hence, the two beams M1A T and M2AT will interfere and produces interference fringes in the field of view of the telescope. This process is known as interference by the division of amplitude.

It should be noted that the beam reflected the mirror M1 passes thrice through G1 while the beam AM2 passes only once through G1. Therefore, a compensating plate G2 of the same thickness as G1 is placed exactly parallel to it to equalize the two paths of the light rays when white light is used as a light source.

When one looks into the beam splitter from the telescope, there appears both the mirror M1 and a virtual image M2’ of M2 formed in G1 which are separated by a distance d = x1 x2. Thus, the two interfering beams reflected by the mirrors M1 and M2 appears as reflected from M1 and M2. Hence, the Michelson interferometer is equivalent to a thin air film enclosed between M1 and M2’, see Figure 10.

The path difference between the reflected beams depends on the separation between M1and M2’ and the inclination of M1 and M2’ respectively. Therefore. the interference fringes may be straight, circular, and parabolic is formed.

Circular fringes:

If the two mirrors M1 and M2 are exactly perpendicular, the reflecting surfaces M1 and M2’ are parallel, so that air film between M1 and M2’ is of constant thickness. Hence, it produces circular fringes of equal inclination corresponding to the different values of θ, sees Figure 11. These fringes are called Haidinger's fringes.

Figure 11: The circular fringe interference pattern produced by a Michelson interferometer

To view these fringes with monochromatic light, the mirrors must be almost perfectly perpendicular to each other. The conditions for constructive and destructive interference pattern can be understood from Figure 12

Figure 12: Virtual images from the two mirrors created by the light source and the beam splitter in the Michelson interferometer

Key Takeaways

2.5 DIFFRACTION: DIFFRACTION OF LIGHT

Diffraction

When the light falls on the obstacle whose size is comparable with the wavelength of light then the light bends around the obstacle and enters the geometrical shadow. This bending of light is called diffraction.

Diffraction is a slight bending of light as it passes around the edge of an object. The amount of bending depends on the relative size of the wavelength of light to the size of the opening. If the opening is much larger than the light's wavelength, the bending will be almost unnoticeable. However, if the two are closer in size or equal, the amount of bending is considerable, and easily seen with the naked eye.

Figure 16: Diffraction of light

In the atmosphere, diffracted light is bent around atmospheric particles -- most commonly, the atmospheric particles are tiny water droplets found in clouds. Diffracted light can produce fringes of light, dark or colored bands. An optical effect that results from the diffraction of light is the silver lining sometimes found around the edges of clouds or coronas surrounding the sun or moon. The illustration above shows how light (from either the sun or the moon) is bent around small droplets in the cloud.

Optical effects resulting from diffraction are produced through the interference of light waves. To visualize this, imagine light waves like water waves. If water waves were incident upon a float residing on the water surface, the float would bounce up and down in response to the incident waves, producing waves of their own. As these waves spread outward in all directions from the float, they interact with other water waves. If the crests of two waves combine, an amplified wave is produced (constructive interference). However, if a crest of one wave and a trough of another wave combine, they cancel each other out to produce no vertical displacement (destructive interference).

Figure 17: Diffraction

This concept also applies to light waves. When sunlight (or moonlight) encounters a cloud droplet, light waves are altered and similarly interact with one another as the water waves described above. If there is constructive interference, (the crests of two light waves combining), the light will appear brighter. If there is destructive interference, (the trough of one light wave meeting the crest of another), the light will either appear darker or disappear entirely.

 Types of Diffraction

There are two types of diffractions

1. Fresnel Diffraction: Fresnel diffraction is produced when light from a point source meets an obstacle, the waves are spherical and the pattern observed is a fringed image of the              object.
    
2. Fraunhofer Diffraction: Fraunhofer diffraction occurs with plane wave-fronts with the object effectively at infinity. The pattern is in a particular direction and is a fringed image of the source.

Figure 18: Types of diffractions Fresnel Diffraction & Fraunhofer Diffraction

From the above figure, we observe that the source is located at a finite distance from the slit, and the screen is also at a finite distance from the slit. The source and the screen are not very far from each other. So this is Fresnel diffraction. Here, if suppose the ray of light comes exactly at the edge of the obstacles, the path of the light is changed. So the light bends a little and meets the screen.

A beam of width α travels a distance of α2/λ, called the Fresnel distance before it starts to spread out due to diffraction. But when the source and the screen are far away from each other, and when the source is located at the infinite position, then the ray of light coming from that infinite source are parallel rays of light. So this is Fraunhofer diffraction.

Here we have to make use of the lens. But why do we use the lens? Because in Fraunhofer diffraction, the source is at infinity so the rays of light that pass through the slit are parallel rays of light.

So to make these rays parallel to focus on the screen, we, make use of the converging lens. The zone which we get in front of the slit in the central maxima. On either side of the central maxima, there is a bright zone i.e. 1st maxima.

Comparison

DIFFRACTION: HUYGEN’S PRINCIPLE

Huygens' Principle

In 1678 Huygens proposed a model where each point on a wavefront may be regarded as a source of waves expanding from that point.

Every point on a wavefront is a source of wavelets that spread out in the forward direction at the same speed as the wave itself. The new wavefront is a line tangent to all of the wavelets.

So Huygens’s Principle states that every point on a wavefront is a source of wavelets, which spread forward at the same speed.

Figure 19: Huygens' Principle

The expanding waves may be demonstrated in a ripple tank by sending plane waves toward a barrier with a small opening. If waves approaching a beach strike a barrier with a small opening, the waves may be seen to expand from the opening.

Huygens-Fresnel Theory:

According to Huygens’s wave theory of light, each progressive wave produces secondary waves, the envelope of which forms the secondary wavefront.

In figure 20 (a). S is a source of monochromatic light and MN is a small aperture. XY is the screen placed in the path of light. AB is the illuminated portion of the screen and above A and below B is the region of the geometrical shadow. Considering MN as the primary wavefront. According to Huygens’s construction, if secondary wavefronts are drawn, one would expect encroachment of light in the geometrical shadow. Thus, the shadows formed by small obstacles are not sharp. This bending of light round the edges of an obstacle or the encroachment of light within the geometrical shadow is known as diffraction. Similarly, if an opaque obstacle MN is placed in the path of light [Figure 20 (b)], there should be illumination in the geometrical shadow region AB also. But the illumination in the geometrical shadow of an obstacle is not commonly observed because the light sources do not point sources and secondly the obstacles used are very large compared to the wavelength of light. If a shadow of an obstacle is cast by an extended source, say a frosted electric bulb, light from every point on the surface of the bulb forms its diffraction pattern (bright and dark diffraction bands) and these overlap such that no single pattern can be identified. The term diffraction is referred to such problems in which one considers the resultant effect produced by a limited portion of a wavefront.

Figure 20: Huygens' Principle

Huygens' principle provides a convenient way to visualize refraction. If points on the wavefront at the boundary of a different medium serve as sources for the propagating light, one can see why the direction of the light propagation changes.

The law of refraction can be explained by applying Huygens’s principle to a wavefront passing from one medium to another (see Figure 21). Each wavelet in the figure was emitted when the wavefront crossed the interface between the media. Since the speed of light is smaller in the second medium, the waves do not travel as far in a given time, and the new wavefront changes direction as shown. This explains why a ray changes direction to become closer to the perpendicular when light slows down.

Figure 21: Law of refraction by applying Huygens’s principle

If we pass light through smaller openings, often called slits, we can use Huygens’s principle to see that light bends as sound does. The bending of a wave around the edges of an opening or an obstacle is called diffraction.

Diffraction is a wave characteristic and occurs for all types of waves. If diffraction is observed for some phenomenon, it is evidence that the phenomenon is a wave. Thus the horizontal diffraction of the laser beam after it passes through slits is evidence that light is a wave.

Diffraction phenomena are part of our common experience. The luminous border that surrounds the profile of a mountain just before the sun rises behind it, the light streaks that one sees while looking at a strong source of light with half-shut eyes and the colored spectra (arranged in the form of a cross) that one sees while viewing a distant source of light through a fine piece of cloth are all examples of diffraction effects. Augustine Jean Fresnel in 1815, combined in a striking manner Huygens' wavelets with the principle of interference and could satisfactorily explain the bending of light round obstacles and also the rectilinear propagation of light.

Key Takeaways

1. When the light falls on the obstacle whose size is comparable with the wavelength of light then the light bends around the obstacle and enters the geometrical shadow. This bending of light is called diffraction.
2. There are two types of diffractions Fresnel Diffraction & Fraunhofer Diffraction
3. Fresnel diffraction is produced when light from a point source meets an obstacle, the waves are spherical and the pattern observed is a fringed image of the object.
4. Fraunhofer diffraction occurs with plane wave-fronts with the object effectively at infinity. The pattern is in a particular direction and is a fringed image of the source.
5. Huygens' Principle states that every point on a wavefront is a source of wavelets that spread out in the forward direction at the same speed as the wave itself. The new wavefront is a line tangent to all of the wavelets.

2.6 THEORY OF PLANE TRANSMISSION GRATING

Key Takeaways

2.7 RESOLVING POWER OF DIFFRACTION GRATING

It is defined as the capacity of a grating to form separate diffraction maxima of two wavelengths that are very close to each other

It is measured by λ/dλ where dλ is the smallest difference in two wavelengths which are just resolvable by grating and λ  is the wavelength of either of them or mean wavelength.

Figure 23: Resolving Power of Diffraction Grating

Key Takeaways

1. It is defined as the capacity of a grating to form separate diffraction maxima of two wavelengths that are very close to each other
2. Resolving Power =    =  nN    

2.8 POLARIZATION

The phenomena of interference and diffraction confirm the wave nature of light hence neither of these phenomena can identify the nature of waves that light is. These phenomena do not tell whether the oscillation of light waves is longitudinal or transverse. This is accomplished by the study of polarization.

A light wave that is vibrating in more than one plane is referred to as unpolarized light.

Example: Light emitted by the sun, by a lamp in the classroom, or by a candle flame is unpolarized light.

This unpolarized light is created by electric charges that vibrate in a variety of directions, this concept of unpolarized light is difficult to visualize. For simplicity, it is helpful to visualize unpolarized light as a wave that has an average of half its vibrations in a horizontal plane and half of its vibrations in a vertical plane. It is possible to transform unpolarized light into polarized light.

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

Figure 24: Polarization

Classification of light

 Light in the form of a plane wave in space is said to be linearly polarized. Light is a transverse electromagnetic wave, but natural light is generally unpolarized, all planes of propagation being equally probable. If the light is composed of two plane waves of equal amplitude by differing in phase by 90°, then the light is said to be circularly polarized. If two plane waves of differing amplitude are related in phase by 90°, or if the relative phase is other than 90° then the light is said to be elliptically polarized.

Linearly Polarized: Light in the form of a plane wave in space is said to be linearly polarized. The electric field of light is limited to a single plane along the direction of propagation.

Circularly Polarized If the light is composed of two plane waves of equal amplitude by differing in phase by 90°, then the light is said to be circularly polarized. if the thumb of your right hand were pointing in the direction of propagation of the light, the electric vector would be rotating in the direction of your fingers.

Elliptically Polarized If two plane waves of differing amplitude are related in phase by 90°, or if the relative phase is other than 90° then the light is said to be elliptically polarized. If the thumb of your right hand were pointing in the direction of propagation of the light, the electric vector would be rotating in the direction of your fingers.

Figure 25: Showing Linear, Circular and Elliptical Polarization

POLARIZATION BY REFLECTION

Production of Polarized Light by Reflection

Brewster’s law can be used to polarise the light by reflection. When unpolarized light is incident on the boundary between two transparent media, the reflected light is polarised with its electric vector perpendicular to the plane of incidence when the refracted and reflected rays make a right angle with each other.

 Figure 26: Polarization by Reflection

Thus we have seen that when the reflected wave is perpendicular to the refracted wave, the reflected wave is polarised. The electric field vectors of all unpolarized light may be resolved into two components as follows;

1. Tangential component- in the plane of incidence.
2. Normal components – in the plane perpendicular to the plane of incidence.

The angle of incidence, in this case, is called Brewster’s angle

 Figure 27: Polarization by Reflection

If the angle of incidence θi is varied, the reflected coefficient ‘r’ for the em waves corresponding to the tangential component will vary as shown in the figure. As θi increases r will decrease.

 At θi = Brewster angle (about 57° for glass) r will be zero and no reflection of the tangential component will take place.

The reflected beam will contain light waves having only the normal component for the electric field as shown in the figure that is the reflected light will be plane-polarized with the oscillations of electric field vector normal to the plane of incidence.

Cause of Polarization by Reflection

The em waves are generated because of the oscillation of electric charge. The electric field vectors of the transmitted ray act on the electrons in the dielectric.

Since the intensity of em wave emitted by the oscillating charge is maximum in a direction normal to the plane of oscillation, and zero along the line of oscillations for the plane-polarized light having electric field vector in the plane of incidence, there will be no emission of em waves in a direction normal to the reflected ray. In the figure, the direction along which no emission should occur is called a zero intensity line. Let the angle between the reflected ray and the zero intensity line be 𝛿  the angle between the reflected and transmitted ray is  π/2 + 𝛿  suppose we arrange the angle of incidence such that

The line of zero intensity and the reflected ray will coincide. Thus there will be no reflection of light when the angle of incidence is equal to Brewster’s angle.

Thus the unpolarized light can be polarized by reflection as the reflected beam will contain light waves having electric field vector oscillating along normal to the plane of the incidence only.

Key Takeaways

1. Polarized light waves are light waves in which the vibrations occur in a single plane. The process of transforming unpolarized light into polarized light is known as polarization.
2. Light in the form of a plane wave in space is said to be linearly polarized.
3. Light is a transverse electromagnetic wave, but natural light is generally unpolarized, all planes of propagation being equally probable.
4. If the light is composed of two plane waves of equal amplitude by differing in phase by 90°, then the light is said to be circularly polarized.
5. If two plane waves of differing amplitude are related in phase by 90°, or if the relative phase is other than 90° then the light is said to be elliptically polarized.
6. Brewster’s law can be used to polarise the light by reflection.

2.9 QUARTER WAVE PLATE AND HALF WAVE PLATE, PRODUCTION, AND DETECTION OF PLANE. CIRCULARLY. ELLIPTICALLY POLARIZED LIGHT

A wave plate or retarder is an optical device that alters the polarization state of a light beam traveling through it. There are many types of retarders. Some common retarders are quarter-wave (λ/4) plates and half-wave(λ/2) plates.

A half-wave plate functions as a polarization rotator for linearly polarized light. It rotates the polarization of a linearly polarized light by twice the angle between its optic axis and the initial direction of polarization, as shown in Figure 28. It introduces a phase difference of radians between the

A typical wave plate is simply a birefringent crystal or a double refracting plastic foil with a carefully chosen thickness. If a beam of parallel light strikes perpendicularly a wave plate the light beam is split into two components due to its double refracting properties. The two components have planes of oscillation perpendicular to each other and slightly different phase velocities.

For a quarter-wave plate, the thickness of the foil is chosen in such a manner that the light component whose electric field vector oscillates in parallel to the rotation lever lags by a λ/4 behind other perpendicular oscillating light components. For a half-wave plate, the thickness is chosen so that the created phase difference has the amount of λ/2. In this experiment monochromatic light falls on a quarter-wave and half-wave plate. The polarization of the emergent light is investigated at different angles between the optic axis of the wave plates and the direction of the incident light.

Figure 28: wave plate

Half-wave plate

A half-wave plate functions as a polarization rotator for linearly polarized light. It rotates the polarization of a linearly polarized light by twice the angle between its optic axis and the initial direction of polarization, as shown in Figure 29. It introduces a phase difference of radians between the two components of the electric field vectors.

 Suppose a plane-polarized wave is normally incident on a wave plate, and the plane of polarization is at an angle ϕ with respect to the fast axis, as shown. After passing through the plate, the original plane wave has been rotated through an angle 2ϕ.

Figure 29: Half-wave plate

A half-wave plate is very handy in rotating the plane of polarization from a polarized laser to any other desired plane (especially if the laser is too large to rotate).  Most large ion lasers are vertically polarized.  To obtain horizontal polarization, simply place a half-wave plate in the beam with its fast (or slow) axis 45° to the vertical.  The l/2 plates can also change left circularly polarized light into right circularly polarized light or vice versa.  The thickness of the half waveplate is such that the phase difference is 1/2 wavelength (l/2, Zero order) or a certain multiple of 1/2-wavelength [(2n+1)l/2, multiple order].

Quarter Wave Plate

A quarter-wave plate is used to convert linear polarization to circular polarization and vice-versa. To obtain circular polarization, the amplitudes of the two E vector components must be equal and their phase difference must be 90 or 270 degrees. A quarter-wave plate is capable of introducing a phase difference of 90 degrees between the two components of the incident light. However, this is not enough to produce circular polarization. The direction of polarization of the incident light with respect to the optic axis of the quarter-wave plate is equally important. It is obvious that the amplitudes of the two components will be equal only when the incident linear polarization direction makes an angle of 45 degrees with respect to the optic axis of the crystal(i.e.α= 45degrees). Figure 30  illustrates the conversion of linearly polarized light into circularly polarized light. Note that conversely, if circularly polarized light is incident on a quarter-wave plate, the resulting polarization will be linear, at an angle of 45 degrees with respect to the optic axis. Again, the reason is that a phase difference of 90 degrees is introduced between the two components of the electric field vector, which traces a helix-like path as the circularly polarized wave propagates.

Figure 30: Quarter Wave Plate

Key Takeaways

1. A wave plate or retarder is an optical device that alters the polarization state of a light beam traveling through it.
2. There are many types of retarders. Some common retarders are quarter-wave (λ/4) plates and half-wave(λ/2) plates.
3. A half-wave plate functions as a polarization rotator for linearly polarized light.
4. A quarter-wave plate is used to convert linear polarization to circular polarization and vice-versa.

2.10 OPTICAL ACTIVITY, SPECIFIC ROTATION

Optical Activity

The optical activity was first observed in quartz crystals in 1811 by a French physicist, François Arago. Louis Pasteur was the first to recognize that optical activity arises from the dissymmetric arrangement of atoms in the crystalline structures or in individual molecules of certain compounds.

Optical activity, the ability of a substance to rotate the plane of polarization of a beam of light that is passed through it. (In plane-polarized light, the vibrations of the electric field are confined to a single plane.

The intensity of optical activity is expressed in terms of a quantity, called specific rotation, defined by an equation that relates the angle through which the plane is rotated, the length of the light path through the sample, and the density of the sample (or its concentration if it is present in a solution). Because the specific rotation depends upon the temperature and upon the wavelength of the light, these quantities also must be specified. The rotation is assigned a positive value if it is clockwise with respect to an observer facing the light source, negative if counterclockwise. A substance with a positive specific rotation is described as dextrorotatory and denoted by the prefix d or (+); one with a negative specific rotation is levorotatory, designated by the prefix l or (-).

Optical activity is the ability of a chiral molecule to rotate the plane of plane-polarised light, measured using a polarimeter.  A simple polarimeter consists of a light source, polarising lens, sample tube, and analyzing lens.

Figure 31: Polarimeter

When light passes through a sample that can rotate plane polarised light, the light appears to dim because it no longer passes straight through the polarising filters.  The amount of rotation is quantified as the number of degrees that the analyzing lens must be rotated by so that it appears as if no dimming of the light has              occurred.
 

Measuring Optical Activity

When rotation is quantified using a polarimeter it is known as an observed rotation, because rotation is affected by path length (l, the time the light travels through a sample) and concentration (c, how much of the sample is present that will rotate the light).  When these effects are eliminated a standard for comparison of all molecules is obtained, the specific rotation, [a].

[a] = 100a / cl    when concentration is expressed as g sample /100ml solution

Enantiomers will rotate the plane of polarisation in exactly equal amounts (same magnitude) but in opposite directions.

As already discussed  Dextrorotary designated as d or (+), clockwise rotation (to the right) Levorotary designated as l or (-), anti-clockwise rotation (to the left).

If only one enantiomer is present a sample is considered to be optically pure.  When a sample consists of a mixture of enantiomers, the effect of each enantiomer cancels out, molecule for molecule.

For example, a 50:50 mixture of two enantiomers of a racemic mixture will not rotate plane polarised light and is optically inactive.  A mixture that contains one enantiomer excess, however, will display a net plane of polarisation in the direction characteristic of the enantiomer that is in excess.

Key Takeaways

1. Optical activity is the ability of a chiral molecule to rotate the plane of plane-polarised light, measured using a polarimeter. 
2. The intensity of optical activity is expressed in terms of a quantity, called specific rotation,
3. specific rotation, [a] is given as [a] = 100a / cl    when concentration is expressed as g sample /100ml solution

2.11 LAURENT HALF’S HALF SHADE POLARIMETER

Laurent’s half shade Polarimeter

Polarimeter – it is an optical instrument used to measure the angle of rotation of plane-polarized light when it is passed through an optically active substance.

By measuring the angle, the specific rotation of an optically active substance can be determined.

Two types of polarimeters are generally used in the laboratory nowadays Laurent’s Half Shade Polarimeter and Biquartz  Polarimeter.  We will discuss Laurent’s Half Shade Polarimeter.

Construction

• S is the source of monochromatic light placed at the focus of convex
• Just after the convex lens, there is a Nicol Prism P is placed. This prism acts as a polariser.
• H is a half shade device that divides the field of polarised light emerging out of the Nicol P into two halves generally of unequal brightness.
• T is a glass tube in which an optically active solution is filled.
• The light after passing through T is allowed to fall on the analyzing Nicol A which can be rotated about the axis of the tube. (Don’t get confused here, we already discussed in the previous section that Nicol prism acts as a polariser as well as an analyzer).
• The rotation of the analyzer can be measured with the help of a scale.

Figure 32: Laurent’s half shade polarimeter

Working

To understand the working of this device we have to know the need for a Laurent half shade device. For this, we assume that the half shade device is not present.

In this basic setup (without the half-shade A) you are looking for the maximum and minimum brightness, which then tells you that the analyzer A is precisely aligned with the output rotation.

When the tube is empty placed the analyzer in such a position that the view of the field is completely dark. Note the position of the analyzer with the help of a circular scale.

Now filled the tube with an optically active solution and it is set in its proper position. The optically active solution rotates the plane of polarization of the light emerging out of the polariser P by some angle. So the light is transmitted by analyzer A and the field of view of the telescope becomes bright.

Now the analyzer is rotated by a finite angle so that the field of view of the telescope again becomes dark. This will happen only when the analyzer is rotated by the same angle by which the plane of polarization of light is rotated by the optically active solution.

The position of the analyzer is again noted. The difference between the two readings will give you the angle of rotation of the plane of polarization.

But here the difficulty associated with this procedure that when the analyzer is rotated for the total darkness, then it is attained gradually and hence it is difficult to find the exact position correctly for which complete darkness is obtained.

Figure 33: Laboratory Set-Up of Polarimeter

To overcome the above difficulty half shade device is introduced between polariser P and glass tube T.

Half Shade Device

• It consists of two semi-circular plates ACB and ADC.
• One-half ACB is made of glass while the other half is made of quartz.
• Both the halves are joined together.
• The quartz is cut parallel to the optic axis.
• The thickness of the quartz is selected in such a way that it introduces a path difference of  ‘A/2 between ordinary and extraordinary rays.
• The thickness of the glass is selected in such a way that it absorbs the same amount of light as is absorbed by quartz half.

Figure 34: Half Shade Device

• Let us consider that the vibration of polarisation is along with OP. On passing through the glass half the vibrations remain along with OP. But on passing through quartz half of these vibrations will split into o  and e-components.
• The e-components are parallel to the optic axis while the o-component is perpendicular to the optic axis.
• The o-component travels faster in quartz and hence an emergence 0-component will be along with OD instead of along OC.
• Thus components OA and OD will combine to form a resultant vibration along OQ which makes the same angle with the optic axis as OP.

  Now if the Principal plane of the analyzing Nicol is parallel to OP then the light will pass through a glass half passable. Hence glass half will be brighter than the quartz half or we can say that the glass half will be bright and the quartz half will be dark. Similarly if the principal plane of analyzing. Nicol is parallel to OQ then quartz half will be bright and glass half will be dark.

  When the principal plane of the analyzer is along AOB then both halves will be equally bright. On the other hand, if the principal plane of the analyzer is along with DOC. then both the halves will be equally dark.

Thus it is clear that if the analysis Nicol is slightly disturbed from DOC then one half becomes brighter than the other. Hence by using a half shade device, one can measure the angle of rotation more accurately.

Mathematically

The specific rotation is given by S=θ/lC where l is the length of the tube and C is concentration.
 

Key Takeaways

1. Polarimeter – it is an optical instrument used to measure the angle of rotation of plane-polarized light when it is passed through an optically active substance.
2. Two types of polarimeters are generally used in the laboratory nowadays Laurent’s Half Shade Polarimeter and Biquartz  Polarimeter. 
3. The specific rotation is given by S=θ/lC where l is the length of the tube and C is concentration.

2.12 PHOTO ELASTICITY

Photoelasticity, the property of some transparent materials, such as glass or plastic, while under stress, to become doubly refracting (i.e., a ray of light will split into two rays at entry).

Optical elements

The method of photoelasticity requires the use of two types of optical elements—the polarizer and the wave plate.

Polarizer: A polarizer is an element that converts randomly polarized light into plane-polarized light. It was the introduction of large polarizing sheets by Polaroid Corporation in 1934 that led to the rapid advance of photoelasticity is a stress-analysis tool.

Before that, small naturally occurring crystals were used for this purpose. In the figure, a single light wave with arbitrarily oriented wave amplitude approaches the polarizer from the left. As this wave encounters the polarizer, it is resolved into two vector components—one parallel to the polarizing direction of the

Figure 35: Polarizer

polarizer, and one perpendicular to it. The parallel component is passed, but the perpendicular one is rejected. The light emanating from the polarizer is therefore plane-polarized in the direction of polarization of the polarizer. Viewed in ordinary (unpolarized) light, a polarizer always looks dark because half the light striking it is rejected.

Wave plate: A wave plate resolves incident light into two components, but instead of rejecting one of these components, it retards it relative to the other component.

Figure 36: Wave plate

In the figure, the “fast” axis of the wave plate makes an angle α with respect to an arbitrarily chosen reference direction. The component f of the incident light with amplitude vector in this orientation is retarded somewhat as it passes through the wave plate. However, the orthogonal component is retarded, even more, resulting in a phase lag δ between this “slow” component and the “fast” one. The term double refraction is often used to describe this behaviour. Waveplates may be either permanent or temporary. A permanent wave plate has a fixed fast-axis orientation α and a fixed relative retardation δ. Such wave plates have their use in photo-elasticity, as will be seen subsequently. A temporary wave plate can produce double refraction in response to mechanical stimulus. Photoelastic specimens are temporary wave plates.

Photoelastic materials are birefringent, that is, they act as temporary wave plates, refracting light differently for different light-amplitude orientations, depending upon the state of stress in the material.

When photoelastic materials are subjected to pressure, internal strains develop that can be observed in polarized light; i.e., light vibrating normally in two planes, which has had one plane of vibration removed by passing through a substance called a polarizer. Two polarizers that are crossed ordinarily do not transmit light, but if a stressed material is placed between them and if the principal axis of the stress is not parallel to this plane of polarization, some light will be transmitted in the form of coloured fringes.

Stresses in opaque mechanical structures can be analyzed by making models in plastic and studying the fringe pattern under polarized light, which may be either white (a mixture of all wavelengths) or a single wavelength.

Advantages and Disadvantages

Advantages

• Photoelasticity, as used for two-dimensional plane problems,
• Provides reliable full-field values of the difference between the principal normal stresses in the plane of the model
• Provides uniquely the value of the non-vanishing principal normal stress along the perimeter(s) of the model, where stresses are generally the largest
• Furnishes full-field values of the principal-stress directions (sometimes called stress trajectories
• It is adaptable to both static and dynamic investigations
• Requires only a modest investment in equipment and materials for ordinary work
• It is fairly simple to use

Disadvantages

On the other hand, photo-elasticity

• Requires that a model of the actual part be made (unless photoelastic coatings are used)
• Requires rather tedious calculations to separate the values of principal stresses at a general interior point
• Can require expensive equipment for precise analysis of large components
• It is very tedious and time-consuming for three-dimensional work.

Key Takeaways

1. Photoelasticity is the property of some transparent materials, such as glass or plastic, while under stress, to become doubly refracting.
2. The method of photoelasticity requires the use of two types of optical elements—the polarizer and the wave plate.
3. A polarizer is an element that converts randomly polarized light into plane-polarized light
4. A wave plate resolves incident light into two components, but instead of rejecting one of these components, it retards it relative to the other component.

Reference Books

I. Engineering physics- Gaur and Gupta, & S.Chand Publication

2. Engineering physics - Avadhanalu and Kshirsagar, S.Chand Publication

3. Fundamentals of optics-Jenkins and White. McGraw Hill Publication

4. A Text Book of Optics Subrahmanyam, BrijIal, S. Chand Publication

5. Engineering physics- Hitendra K Malik, A.K.Singh, Tata McGraw Hill Education Private Limited, New Delhi

6 Essential University Physics Volume – l and 2 Richard Wolfson, Pearson, Noida 7. Engineering Physics Dattu R Joshi Tata Mc-Graw Hill Education Private Limited