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Polarization

Electromagnetic Spectrum:

The areas of Optical sciences normally refer to a small spectral region of electromagnetic

radiation. Normally it covers the region from UV to IR region. Every part of electromagnetic

spectrum has one or more than one application.

Electromagnetic wave as the transverse wave:

It is well known fact that the electromagnetic radiation behaves as the transverse wave.

Thus the light behaves as transverse waves. This means that the electric and magnetic field

disturbances are in the plane transverse to the direction of propagation of the wave. Electric and

magnetic fields are also perpendicular to each other and are interrelated also.

Essential parameters of Electromagnetic wave:

For complete mathematical representation of the electromagnetic wave it is sufficient to know

the following:

(a) Frequency of the wave or angular frequency : A fundamental property of the wave, so

long it is propagating through a linear medium.

(b) Amplitude and the direction of electric field E0 :

(c) Propagation vector k whose magnitude is (c being the velocity of propagation of

the em wave and its direction represents the direction of propagation of the em wave).

For a pure monochromatic wave moving along the z-axis (also called as the longitudinal

direction), the electric field can be written as

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or simply .............(1.1)

Fig1

Thus at any given time t , the variation of the electric field along the longitudinal direction issinusoidal as shown in fig 1(a). The variation of electric field at a particular location z as a

function of time t is also sinusoidal as shown in fig 1(b).

The relation between electric field and the magnetic field for an electromagnetic wave is given

by

..... (1.2)

Thus equation 1.1 is a complete description of the electromagnetic wave described with the help

of three parameters defined, above

(d) Phase Factor: The factor is termed as the phase factor and plays an important role in

determining the status of the wave at any instant of time and or at any location in space. If the

phase factor is constant in the transverse plane x-y, than the wave will be called as the planewave and all the rays in it will be parallel to each other. The planes containing the constant phase

are termed as the wavefront. The wave front is perpendicular to the direction of propagation. Fig

2 shows the plane wave front for a plane wave propagating along z axis.

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(e) Intensity: The power (energy) associated with the radiation field is measured in terms ofintensity I. It is defined in terms of watt/cm

2. It is related to the electric field associated with em

wave given by

........................... (1.3)

where is the velocity of propagation of the wave and is the electrical permittivity of the

medium through which the wave is propagating.

(f) Propagation of EM wave: Electromagnetic waves obey the Maxwell’s equations.

While talking about the propagation or the transmission of light through a medium without

undergoing significant attenuation, then the medium is a dielectric. The Maxwell’s equations for

a dielectric medium are given below:

.................. (1.4)

For a linear medium, the displacement vector and (where is the permittivity

of the medium).

The equation 1. 4 results into the wave propagation equation given by

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............................ (1.5)

From equation 1.5, the velocity of propagation of em wave is given by

.................... (1.6)

If the wave is propagating in vacuum then the free space permittivity and permeability are given

by =8.85X10-12

s2 .C

2/m

2 Kg and =4 X10

-7 m.kg/C

2 then the speed of light in free space is

3X10 8 m/sec, this normally denoted by c and not v.

Therefore c=3X108

m/sec. The ratio of the speed of light in vacuum to that of in medium iscalled as the refractive index of the medium n,

............................. (1.7)

Refractive index of the medium plays an important role in determining the polarization of light

through that medium.

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1. Electromagnetic waves at the interface of two dielectric mediums.

When an electromagnetic wave strikes at and air dielectric interface, from air to the dielectricmedium, part of it is reflected back and part of it is transmitted into the dielectric medium. The

amount of light reflected and transmitted depends on the angle of incidence, refractive indices of

the medium forming the interface and the angle of incidence.

Let us consider an interface formed by two dielectric medium I and II of refractive index n1 and

n2 respectively as shown in fig 1.In the previous lecture we summed up the properties ofelectromagnetic waves and its representation. In this lecture we will be looking into the

modification which an EM wave under goes when it strikes the interface of two dielectric

medium.

fig 1: The different portions of light at a material interface

A ray of light from the medium 1 incident on the interface making an angle of incidence with

normal partially gets reflected at the angle of reflection being equal to the angle of incidence and

partially refracted at an angle of incidence fowling the Snell's law given by

.

2. Reflection and the transmission coefficient as a function of angle of incidence

The amount of light reflected and refracted or transmitted depends on the optical property of the

medium viz; refractive index, direction of electric field and the angle of incidence.

The direction of electric field is always perpendicular to the direction of propagation of the wave.

If a wave is traveling in a certain plane the electric field vector could be in two mutually

perpendicular planes which are normal to the direction of propagation. For example if the wave

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is propagating along z direction the electric field can be in the z-x (or z-y) plane or in x-y plane

(or could be in both these planes; randomly polarized) but always perpendicular to the direction

of propagation.

Similarly in the present case of interface of two dielectric media, two mutually perpendicular

planes can be defined as

1. The plane of the incidence containing

the incidence ray, the reflected ray and

the normal to the interface as shown infigure 2 and

2. The plane perpendicular to the plane of

incidence as shown in fig.

If the electric field is only in the plane of

incidence then the wave is called as p polarizedand if the electric field is perpendicular to the

plane of incidence then the wave is called as s

polarized.

fig : 2

2.1 Fresnel's equations

Fresnel's equation relates the ratio of the amplitude of the electric field of reflected and

transmitted light to that of the incident field, angle of incidence and the refractive indices of twomedia forming the interface. Let us denote the amplitude of incident electric field in the plane of

incidence as and that of nomal to the plane of incidence as . Accordingly the reflected and

transmitted amplitude of electric fields are:

Amplitude of the parallel component of electric field transmitted.

Amplitude of the parallel component of electric field transmitted.

Amplitude of the normal component of electric field reflected.

Amplitude of the normal component of electric field transmitted.

The ratio of the reflected amplitude to that of the incidence amplitude is called as amplitude

reflected coefficient r ( or , for corresponding to parallel or normal component respectively)

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and that of for transmitted component is called as amplitude transmitted coefficient ( and ).

The expressions for these coefficients are given by

.............................. (1)

............................... (2)

............................... (3)

................................ (4)

These four equations are called as Fresnel's equations and can be obtained by solving the wave propagation equations under the appropriate boundary conditions ((Ref: D.J. Griffiths,

Introduction to Electromagnetics, Prentice Hall (1995). ).

The ratio of corresponding reflected (transmitted) intensity to the incidence intensity is called as

the reflectivity (transmittance) for the interface.

Thus,

1. Brewster’s angle: The Fresnel's equations eqs 1-4 tell us the variation of amplitude

coefficient fro reflected and the transmitted ray for the given interface. The amplitude reflection

Coefficient for the air –glass interface as a function of angle of incidence is shown in fig 3(a) for

both the s polarized (perpendicular component) as well as p polarized (parallel component)wave. The corresponding reflectivity is shown in fig 3(b). Here air is taken as the first medium

with refractive index n 1 =1 and the glass as the second medium having refractive index n = 1.5 .

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Fig 3. (a) Variation of amplitude coefficient of reflection as a function of

(b) Reflectivity vs. a function of angle of incidence plotted.

The variation of amplitude coefficient of transmission as a function of angle of incidence is

shown in fi. 4(a). Corresponding transmittance of both the waves shown in fig 4(b).

Fig. 4 (a) Variation of amplitude coefficient of transmission as a function of angle of

incidence.

(b) Variation of transmittance of both the waves is shown.

Equation 1 shows that in the present case where n1 < n2 (air glass interface), the amplitude

coefficient of reflection for the s wave (perpendicular component) is negative, indicating that the

reflected and the incident wave are out of phase. The magnitude of reflection coefficient for this

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particular components increases with the increasing angle of incidence going to the value of 1 at

(fig 3) .

Using the snall's law, equation 3 can be re-written as

..........................(5)

This will be positive and decrease with the angle of incidence for

or equivalently

......................................(6)

For the present case of air-glass interface, where n1 < n2 , the above inequality will hold for

..........................................(7)

For ...........................................(8)

For , indicating that the reflected light will only be only s polarized. Beyond this

............................................(9)

will be negative and its magnitude will increase with the angle of incidence taking the value of

1 at as shown in fig 3. the negative sign in the region of inequality (7) indicate that the

electric field for the reflected wave is out of phase with the corresponding incident wave.

For the angle of incidence satisfying equation (8), the reflected light is purely s polarized. This

particular angle of incidence denoted by is termed as the Brewster angle and from equation

(8) it can be easily shown .

................................................(10)

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In case if the first medium happen to be other than air than than equation (10) can be shown to be

modified as

Thus if an electromagnetic wave falls at an Brewster's angle then the reflected light is S

polorized. Hence this is one of the way to generate the polorized light.

Randomly polarized light

In the previous lecture, we have realized that the electric field associated with a plane

electromagnetic wave can be in two mutually perpendicular plane (maintaining the direction ofelectric field perpendicular to the direction of propagation). If a source is emitting the radiation

having the electric field in two mutually perpendicular plane simultaneously then such radiation

is termed as randomly polarized light. Or in other words if both, s polarized and p polarized

wave, are present simultaneously the wave is randomly polarized light .

Linearly or Plane polarized light:

Let us consider a plane electromagnetic propagating along z axis whose electric field can be

expressed by

............................(1)

In this case, the electric field will be sinusoidal as a function of time for a given location,z, in the

path of the propagation or for a given time, the electric field will go sinusodially as function of zas shown in figure1. But the direction of electric field at any time and at any point in space is

always same i.e along the x-axis for the representation of eq 1.

Similarly we can have a have an plane wave propagating along the z-axis having the electric

field along the y axis given by

.............................. (2)

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There are two differences between the wave represented by eq.(2) with that of eq.(1). There is a

phase difference between the two waves and the direction of the electric filed is always alongy axis for eq.(2).

Fig: 1.

Suppose that the frame of reference is rotated about z axis by an angle in anticlockwise

direction as shown in fig.2

Fig: 2

Let us denote the new corresponding axis as x' and y' and their unit vectors as and

respectively. In this new frame of reference the electric field associated with the electromagnetic

wave represented by by eq 1.

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or,

..........(3)

where represents the component of the electric field along x' andthat of along y' for the eq 3 (which is representing the same field of eq 1).

Electric field for eq 2 in the new coordinate system

or .....................(4)

where represents the component of the electric field along x' and

that of along y' for the eq 4 (which is representing the same field of eq 2).

In all the above representation , the direction of electric field was always fixed, ie it is neither

changing with time nor changing with the location and amplitude of the wave is also constant.

Such waves are called as linearly polarized or plane polarized wave.

From equation 3 or from equation4, a general form of linearly polarized light is

or .............................. (5)

Where the resultant amplitude of the wave is and the direction of electric field

is always making an angle with x-axis (as shown in fig 3 a) given by .

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Figure 3.

Representation of polarized light in Jones calculus: Sometimes, for convenience, equation(5)

can also be written in terms of complex exponential form

................................ (6)

quation (6) can be written in terms of column vector notation

...............................(7)

The column vector, in eq 7 is called as Jones vector J of light wave.

In this notation, jones vector for a light polarized along x axis only is written as

...............................................................(8)

and the light polarized along y axis only is given by

................................................................(9)

A linearly polarized light making an angle of 45 with x-axis can be written as ( )

..................................................................(10)

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Superposition of two different polarized wave :

Consider a situation where light field given by eq 1 and eq2 both are present,

Then the resultant field is given by

......(11)

If then above equation is

....................................................(12)

If than above equation (11) is

.......................................................(13)

Eq. (12) and eq. (13) represent a plane polarized light. Thus if two plane (or linearly) polarizedwaves of same frequency and having a phase difference in the multiple of p are superimposed,

they result into a plane polarized wave only.

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Polarization of light by reflection

Malus law

In lecture 3 and 4, module 13, we have studied various categories of polarized light.

In this module, the question of how to obtainted polarized light or how to change the poarization

of the wave. is discussed

1. Polarization of light by reflection

1.1 Polarization at Brewster angle

We have seen in the previous module that the reflectivity of an EM wave on an interface formed

by two dielectric medium depends on the refractive index of both the medium, angle of incidence

and the polarization of the wave. It is also shown in lecture 2, module 13, section 2 when the

angle of incidence is equal to the Brewster angle, then the reflectivity for the p

component ( ) is zero. Thus if a randomly polarized light is made to incident at Brewster angle

on air-glass interface then the reflected light will be s polarized (Fig 1.).

Fig 1: Generation of polarized light at Brewster's angle.

1.2 Change of Polarization by Internal Reflection

Let us consider an interface formed by two dielectric medium as shown in Fig 2 (a),

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light is incident from medium 1 having refractive index n 1 to the medium of refractive index n 2

, such that n 1 >n 2 .Recalling the snell's law eq 1 from lecture 2, module 13, the angle oftransmission will keep on increasing as the angle of incidence increases. At certain angle of

incidence , the angle of refraction will be 90 ° given by the condition.

--------------------------(1)

for the angle

Which is not possible

At , refracted ray will go tangentially as shown in Fig 2(b). Beyond this angle of incidence i.e.

the transmitted ray will not exist. The complete radiation will be reflected back in thesame medium, Fig 2(c). This phenomena is termed as total internal reflection. The minimum

angle at which this phenomena occurs is termed as the critical angle .

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On analyzing the fresnel's equation, 1-4, lecture 2, module 13, one can clearly find out that the electric

field becomes complex beyond the critical angle thus adding a phase change on reflection. For ,

is imaginary, on solving eq 1 and 3 (lecture 2, module 13, the phase changes on reflection can be

found for both the s and p components given by (assuming n 2 =1, air)

----------------------(2)

----------------- (3)

where and are the phase shift on reflection (for ) for s and p components

respectively.

Fig 3 shows the variation of and as a function angle of incidence .

Fig 3; Variation of Phase change beyond the critical angle.

A specific polarized light falling beyond a critical angle, will undergo a change in polarization

because of relative changes in the phases after reflection for s and p components. For example, a

linearly polarized light may become an elliptically polarized light depending upon the phase

difference and amplitude of s and p components. As an example, let us consider a

plane polarized light falling on glass. (n=1.51) –air interface at an angle of 55 (> ) on glass asshown in figure 4.

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Fig 4: Change in polarization of light as a result of total internal reflection.

The phase changes on reflection are 58.6 and 103 for s and p polarized light respectively.

Therefore the phase difference .

2. Polarization of light by pile of plates.

The transmitted wave at the Brewster angle contains the p polarized light and the s polarised

light for which the electric field and hence intensity given by eqs 2 and 4, lecture 2, module 13.

The transmitted light is randomly polarized (at Brewster angle) or can be termed as partially

polarized light. The transmitted light can be converted into p polarized light completely byallowing it to pass through large number of interfaces similar to that of fig 1. One can use a large

number of glass plates separated by a small distances, in air, to form a series of air- glass

interface as shown in fig 5.

Fig 5: Polarization of transmitted light by pile of glass plates.

In this case, if the light is falling at Brewster angle then at each interface there will be thereflected light containing only s component and so from the transmitted light the s component

will vanish slowly and leaving only the p component. One can define the degree of polarization

P for the transmitted component as

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-----------------------------------(4)

I s and I p are the intensities of s and p component in the transmitted light respectively. If there

are m glass plates placed at Brewster's angle then the degree of polarization is given by

-----------------------------------(5)

Where, m is the number of glass plates used and n is the refractive index of the glass plate. From

eq 5 one can say that if m is very large then degree of polarization for the transmitted light will

be approaching to one, thereby the transmitted light will be completely p polarized. Therefore the

pile of glass plates at Brewster angle, gives only p polarized light in the transmitted direction.

Thus by carefully choosing the interface and the angle of incidence (or the optical geometry ofthe system) one can get s or p polarized light.

The pile of such glass plate can be termed as polarizer, which has a property to deliver the

polarized light.

3. Malus Law

We have seen above that a pile of glass plate acts as a polarizer. Suppose that Two such pile of

plates are used as shown in fig 6.When the planes of incidence for both the sets are parallel (i.e

angle of incidences are at Brewster angle) the output from the second plate will be p polarized

light. . Now suppose that the second pile of plate is rotated about the axis X-X'

as shown suchthat the plane of incidence changes for the second set of glass plates.. You will observe that the

light intensity will start decreasing and finally when the second plate is rotated by 90 ° , the

output light is zero because for this position the electric field is perpendicular to the plane ofincidence and the light will be reflected completely from it. On further rotation, the light output

will be restore at 180 ° . The cycle will continue from 180 ° -360 ° . Thus in one complete

revolution of the second pile the light will under go maxima and minima twice.

Fig (6)

We can say that first pile of glass plate is acting as a polarizer and the second pile is analysisng it

and hence can be termed as analyzer. The orientation of the analyzer which allows the polarized

light to pass completely is termed as the pass plane of the analyzer.

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Let us understand quantitatively the variation of light intensity as a function of the rotation angle

of analyzer.

The electric field E of the light coming out of the polarizer (polarized light) can be resolved in

two mutually perpendicular components. E1 component, having electric field in the plane of the

incidence of analyzer and other E2 perpendicular to it as shown in Fig. 7.

Fig 7: Components of electric filed in two mutually perpendicular direction.

The component E1 will pass through the analyzer completely. Thus if the plane of the polarizer is

making an angle of with the pass plane of the analyzer then the amplitude of the light passing

through the analyzer is

----------------------------------------(6)

and corresponding intensity

--------------------------(7)

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where I 1 is the intensity of light coming out of analyzer and I is that of incident on it. Equation 7

is known as the Malus law. The plot of I 1 verses the orientation angle is shown in fig 8.

The pile of glass plates can be replaced by certain special type of material called as polarizingmaterial (or Polaroid) which allows only specific polarized light to pass completely (neglecting

the absorption). A randomly polarized light falling on to the polarizer will allow to transmit only

one particular component of electric filed (define by the pass plane of the polarizer) thus

generating the plane polarized light. A second polarizer called as analyzer can detect it. Malus

law given by eq (7) will hole; in this case q will be the angle between the pass plane of polarizer

and analyzer.

Phenomena of double refraction

In previous lectures, so far we studied the cases where the propagation of light through adielectric medium is independent of its state of polarization and direction of propagation. That

means that a randomly polarized light or two mutually orthogonal polarized lights will see the

same refractive index in all the direction while propagating through the medium. Recalling theHuygen's principle, one can say that if there is a point source of light inside the medium, it will

emit a spherical wave front, as shown in Fig.1, irrespective of the state of polarization. Such

materials are called as isotropic medium.

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Ordinary glass is a good example of isotropic material. There are certain material for which the

refractive index is different in different direction or we can say simply that the speed of

propagation will not be same in all direction and also it may depend on the state of polarization.In such materials a point source may result into the elliptical wave warfront or for different

polarization the wave front will be different, as shown in fig 1(b). Such materials are called as

anisotropic material. Crystals like Quartz (SiO 2 ), calcite (CaCo 3 ) and KDP (potassiumdihydrogen phosphate, KH 2 PO 4 ) are good example of anisotropic material. Such anisotropy is

due to difference in the arrangement of the atoms and molecules in different direction in the

crystal which leads to direction dependent polarization property of the system. Recall thatrefractive index n is given by (module 13, lecture 1, eq 1.7)

Which depends on permitivity or polarization property of the medium. The polarizibility of a

medium is given by for isotropic medium,

,

implying that permitivity is independent of the direction. For anisotropic medium, in

the simplest possible case,

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,

and

.

From this, one can say that the refractive indicies are different in different directions for

anisotropic crystals and hence the speed of propagation of light in such crystals will depend on

the direction of propagation and the polarization of light. These crystals are called as birefringent

crystals or doubly refracting medium.

When an unpolarized light (which is equivalent to the two orthogonally polarized light) falls onsuch crystals, different polarization will propagate through the crystal with different speed and it

will result into elliptical wave front as shown in Fig 2.

In general a crystal may exhibit three principal refrective indixcies corresponding to threemutually orthogonal axis. In some of the crystals, there may be one particular axis, say z axis

such that if the light is propagating along this axis then the two mutually orthogonal waves may

be same. Such axis is called as the optics axis and the crystal is termed as uniaxial crystal. In auniaxial crystal, the propagation of light along the optics axis or along the z axis is independentof the state of polarization, just like the isotropic system. The refractive index in such situation is

denoted as no. In case if the light is traveling along x (or y) direction, other than optics axis then

the refractive indices will be different for two mutually orthogonal polarization. The light polarized along y direction (or x) will have refractive index no and that of along z direction it is

denoted by ne accordingly the waves are called as ordinary and extraordinary rays.. Suffix e

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denotes the extraordinary behaviour. In other words, one can say the uniaxial crystals offers two

refractive indices no and ne and hence are birefringent material.

• Propagation of light through birefringent crystal :

Let us consider a uniaxial crystal placed in air. As discussed in the previous module, theradiation falling at an interface may be resolved in two kinds of polarization, viz: in plane

polarization and perpendicular polarization. To make the discussion simple, we will consider the

normal incidence only. If no> ne, the crystal is called as positive crystal and if no< ne, the crystal is

called as negative crystal.

Case I:Let us consider the first case when the optics axis is perpendicular to the interface formed

by the uniaxial crystal and the air as shown in Fig.2. In this case, the refractive indicies for both

polarization is same and is given by no.

Both the s and p waves will travel with the same speed. One can show the wave front by drawing

a circile of radius c/no with the center at the interface at point O, where the radiation is hitting theinterface as shown in fig. The wave front will be spherical corresponding to both the

components, . The situation will be similar to that of isotropic medium.

Fig.3 : Propagation of wave in uniaxial

crystal when the optics axis is oriented

along the direction of propagation

Case II . In the second case, the optics axis is parallel to the interface as shown in Fig 4.

For the s wave (having polarization perpendicular to the plane of incidence or to the plane of

the paper) the refractive index is no (and hence termed as ordinary wave) and the wave front will be spherical wave front as shown in fig 4. But for p wave the refractive index is ne (or it can betermed as extraordinary wave)and the wave front will be elliptical wave front. It can be drawn by

plotting an ellipse with the major axis as c/ne along the direction of propagation and the minor

axis as c/no along the optics axis. In this case, s and p wave will develop a relative phase

difference while propagating the same length (physical ) in the crystal owing to different

refractive indices.

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Fig 4. Prpoagation of plane wave in uniaxial crystal when optics axis is oriented parallel to

the plane of the paper and perpendicular to the direcytion of propagation.

Case III . In this case the optics axis is perpendicular to the plane of the page. Optics axis is still

parallel to the interface as in the case II. So in this case only difference is that s polarization iswill see a refractive index ne and it will extraordinary wave and p polarization will observe

refractive index no and will be ordinary ray as shown in fig 5.

Fig 5. Propagation of plane wave in uniaxial crystal when optics axis is oriented

perpendicular to the plane of paper as well as perpendicular to the direction of

propagation.

Case IV In this final case let us consider the optics axis to be in the plane of incidence but is

along the direction as shown by the broken lines in fig 6. let us consider a plane wave AB

incident normally on such interface. The s wave will propagate normal with the refractive indexno and the wave front corresponding to this polarization will remain un deviated inside the

crystal as shown by OO' . This is the ordinary ray But for the in plane p wave the wave front is

elliptical as shown in the fig. 6. This wave front can be obtained by plotting an ellipse having

center at C or D with major axis to the optics axis having magnitude c/ ne and the minor axis

as c/no . It is obvious from this Hugenn's construction that the p wave will under go a deviationand corresponding wave front is denoted by EE'. Hence under such case the two waves will split

into two components even under normal incidence, one of them following the undeviated path

termed as ordinary wave and the other under going deviation is termed as extraordinary ray.

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Next lecture will demonstrate how to use such properties of uniaxial crystal to generate the wave

of required polarization froma plane polarized light.

Fig.6 : Propagation of plane wave in uniaxial crystal when optics axis is in the plane of

incidence but oriented at certain angle with respect to the interface.

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