what minimum thickness should a piece of quartz have to acts as a quarter wave

Representation of polarized and unpolarized light

Let usa consider an ordinary light ray passing perpendicular to the plane of the paper and into the paper. The electric field vectors are perpendicular to the ray propagating with equal aamplitude in all possible directions as shown in Fig. 14.22(a). This is the nature of unpolarized light.

The linearly polarized low-cal is shown in Fig. xiv.22(b) and in 14.22(c). In Fig. 14.22(b), the direction of electrical field vectors lie in the plane of the paper and in Fig. 14.22(c) the management of electric field vectors are perpendicular to the airplane of the paper. The positions of perpendicular vectors of the ray are shown with dots.

Effigy 14.22 (a) Electrical field vectors of unpolarized low-cal; (b) Vertically airplane polarized light; (c) Horizontally plane polarized low-cal; (d) Unpolarized light

In a light ray if the electrical field vectors perpendicular to the management of ray propagation do non have equal amplitudes, and so the field vectors have been resolved along ii perpendicular directions, sayten andy directions. If the resultant electrical field amplitudes forthten andy directions are equal, then the ray is said to be unpolarized ray. On the other hand if they are not equal then the ray is said to be polarized along the larger resultant field amplitude. Unpolarized light is represented as a combination of that shown in Figs. 14.22b and 14.22c in Fig. fourteen.22d.

Polarization by reflection

In 1809, Malus, a French scientist discovered that when ordinary calorie-free in incident on the surface of a transparent medium similar glass or water, then light tin be partially or completely polarized on reflection. The extent of polarization of reflected light varies with the angle of incidence. In 1811, Sir David Brewster noticed the extent of variation polarization of reflected lite by varying the angle of incidence on the surfaces of unlike transparent materials. He observed that for a particular angle of incidence [θ _i =θ _p,θ _p = angle of polarization] the reflected light is completely plane polarized equally shown in Fig. 14.23. This angle of incidence is known as Brewster's angle or angle of polarization. The angle of polarization varies with material also. The Brewster's angle for drinking glass (μ = i.52) is 57°.

Suppose the incident beam makes Brewster's bending, and then the reflected light is completely plane polarized while the transmitted calorie-free is partially plane polarized is shown in Fig. xiv.23. In the reflected lite the vibrations of the electric vectors are perpendicular to the aeroplane of incidence. The plane of incidence is the aeroplane containing incident beam and the normal to the surface at the point of incidence. The intensity of reflected wave is less and it is nearly xv% intense as compared to the intensity of incident beam, while the intensity of transmitted beam is large and is partially polarized. By using a large number of thin parallel drinking glass plates instead of single drinking glass plate as shown in Fig. 14.24, the intensity of reflected waves is enhanced and the transmitted beam becomes more aeroplane polarized. The vibrations of electric vectors in the transmitted beam is in the plane of incidence.

Effigy 14.23 Polarization past reflection

igure 14.24 Polarization of light past a stack of glass plates

Experimentally it was constitute by Brewster that when the angle of incidence (θ _i) is equal to the polarizing bending (θ _p) the angle betwixt the reflected ray and refracted ray is 90°.

Henceθ _p + 90° +θ _r = 180° [Since angle of incidence = angle of reflection andθ _r= bending of refraction]

orθ _p +_{θ r} = 90° _________ (14.55)

From Snell's law

northward ₁ sinθ _p =n _ii sinθ _r _________ (14.56)

[n _i = refractive alphabetize of air andn ₂ = refractive index of glass plate]

Using equation (14.55) in equation (fourteen.56),

The to a higher place equation is known equally Brewster's law, because Brewster deduced it emperically in 1812.

Malus police

Plane polarized calorie-free is obtained by passing unpolarized light through a polarizer. When airplane polarized lite from the polarizer is passed through the analyser, the intensity of polarised light transmitted through the analyser varies equally the foursquare of the cosine of the angle between the airplane of transmission of the analyser and the airplane of the polarizer. This is known as Malus constabulary.

Malus police can be proved by considering the aamplitude (a) of the incident plane polarized low-cal on the surface of the analyser every bit shown in Fig. fourteen.25. Letθ be the angle betwixt the planes of the analyser and the polariser.

The amplitude of incident aeroplane polarised calorie-free parallel to the plane of transmission of the analyser isacosθ and perpendicular to information technology isasinθ.

Only the parallel component pass through the analyser.

Figure 14.25 Malus law

∴ The intensity of light transmitted through the analyser is

I = (a cosθ)² =a ^two cos²θ _________ (14.58)

If I₀ =a ² is the intensity of plane polarized light on the surface of the analyser and so equation (14.58) becomes

I = I₀ cosⁱⁱ θ _________ (fourteen.59)

Whenθ = 0, I = I₀ and ifθ = xc° then I = 0. The above results have been proved for tourmaline crystals, nicol prisms, etc.

Double refraction

When a beam of unpolarised calorie-free passes through anisotropic crystals such equally quartz or calcite, the beam will separate into two refracted beams. This is known equally double refraction or birefringence. The direction in which the ray of transmitted light does non suffer double refraction inside the cystal is known as the optic axis. If only one optic centrality is present in a crystal then it is called uni-axial crystal. On the other hand if two optic axes are nowadays in a crystal then it is known as biaxial crystals.

Double refraction in calcite crystal is shown in Fig. 14.26. The refracted beams are aeroplane polarised. One beam is polarised along the direction of the optic axis and is known as extraordinary ray (e-ray), while the other refracted beam is polarised along the direction perpendicular to the optic axis and is known every bit ordinary ray (o-ray).

Figure 14.26 Double refraction in calcite crystal

The wave fronts of e-ray and o-ray are shown for quartz and calcite crystals in Fig. 14.27.

The velocity of o-ray is the same in all directions of a crystal, whereas the boggling ray travels with different velocities in unlike directions. The velocities of o-ray and east-ray are the same along the optic axis. The velocity of e-ray is less than that of o-ray in quartz crystal, this is a positive crystal. The velocity of e-ray is more than than that of o-ray in calcite crystal, this is a negative crystal.

Figure 14.27 (a) Wave fronts in quartz crystal; (b) Moving ridge fronts in calcite crystal

Forth the optic centrality, the refractive alphabetize μ _eastward  = μ ₀ . So the wave fronts of e-ray and o-ray volition coincide. In quartz crystal μ _eastward  > μ _o  in all other directions, and it is very big in the management perpendicular to the optic axis. Calcite crystallises in rhombohedral (trigonal) crystal organisation. The diagonal line that passes through the blunt corners is the optic axis. In calcite μ _e  < μ _o  in all directions except the optic centrality. So, the speed of eastward-ray is larger than that of o-ray. μ _e  is very much less in the direction perpendicular to the optic centrality.

Nicol prism

Nicol prism is an optical device used to produce and analyse airplane polarized light. This was invented by William Nicol in the year 1828. Nicol prism is fabricated from a double refracting calcite crystal. As shown in Fig. 14.28, a calcite crystal whose length is three times its breath is taken. The corners A′Thou′ is blunt and A′C Thou′E is the chief department with ∠A′CG′ = 71°. The finish faces A′BCD and EFG′H are grounded and then that the bending ACG = 68°. The crystal is cut along the plane AKGL. The cutting surfaces are polished untill they are optically flat and cemented together with Canada balsam.

The refractive index of Canada balsam (1.55) is in betwixt the refractive indices of ordinary (1.658) and extraordinary (ane.486) rays in calcite crystal.

The section ACGE of Fig. 14.28 is shown separately in Fig. 14.29. In Fig. 14.29, the diagonal AG represents the Canada balsam layer. A beam of unpolarized calorie-free is incident parallel to the lower edge on the face ABCD. They are doubly refracted on inbound into the crystal. From the refractive alphabetize values, we know that the Canada balsam acts every bit a rarer medium for the ordinary ray and it acts every bit a denser medium for boggling ray.

When the angle of incidence for ordinary ray on the Canada balsam is greater than the critical angle then total internal reflection takes identify, while the exaordinary ray gets transmitted through the prism.

Nicol prism tin be used as an analyzer. This is shown in Fig. 14.30.

Effigy fourteen.28 Calcite crystal

Figure fourteen.29 Production of plane polarized lite using Nicol prism

Fig. fourteen.xxx (a) Parallel Nicols; (b) Crossed Nicols

Two nicol prisms are placed adjacently as shown inFig. xiv.xxx(a). Ane prism acts as a polarizer and the other act equally an analyzer. The exaordinary ray passes through both the prisms. If the second prism is slowly rotated, and so the intensity of the exaordinary ray decreases. When they are crossed, no lite come out of the second prism because the e-ray that comes out from first prism will enter into the 2d prism and deed as an ordinary ray. Then, this low-cal is reflected in the second prism. The first prism is the polarizer and the second one is the analyser.

 Quater-wave plate

A quater-moving ridge plate is a thin double refracting crystal of calcite or quartz, cut and polished parallel to its optic axis to a thickness 'd' such that it produces a path difference of or phase difference of betwixt the o-ray and east-ray when plane polarized calorie-free incident normally on the surface and passes through the plate.

Every bit shown in Fig. 14.31, consider a calcite crystal of thickness 'd'. The optic axis is parallel to the surface. When a plane polorized light is incident unremarkably on the surface, so the light will split upwardly into o-ray and e-ray. These rays travel with different speeds in the crystal. In calcite crystal the e-ray travel faster than o-ray. Hence the refractive index of o-ray (μ _o) is higher than the refractive alphabetize of eastward-ray (μ _e) in the crystal. The optical path covered past the o-ray as it pass through the crystal of thickness 'd' isμ _od. Similary the optical path covered by the east-ray as it pass through the crystal of thickness 'd' isμ _ed.

∴ The path difference, Δ =μ _od –μ _ed = d(μ _o –μ _e)

Equally the crystal is a quater-wave plate, information technology introduces a path difference of between o-ray and e-ray.

Figure 14.31 Propagation of polarized low-cal in calcite quater-wave plate

Therefore Δ =

Then nosotros can write,

For another crystals like quartz, whereμ _eastward >μ _o, the thickness of the quarter moving ridge plate Using the above equation, the thickness of quarter wave plate can exist estimated.

Half-moving ridge plate

Similar to quater-wave plate, a half-moving ridge plate introduces a path divergence of or phase departure of π betwixt o-ray and e-ray. Let 'd ' be the thickness of the half-wave plate.

For a half-moving ridge plate, the path difference, in calcite crystal half-wave plate. In the example of quatz,. Using the in a higher place equation, the thickness of one-half-wave plate can be estimated.

Theory of round and elliptically polarized light

A beam of plane polarized lite tin can be obtained from a Nicol prism. This beam of plane polarized light is made incident normally on the surface of a calcite crystal cutting parallel to its optic axis.

Every bit shown in Fig. fourteen.32(a), let the plane of polarization of the incident beam make an bendingθ with the optic centrality and let the amplitude of this incident light beA.

Figure 14.32 (a) Plane wave incident on calcite crystal; (b) east-ray and o-ray light amplitudes in calcite crystal

As polarized lite enters into the calcite crystal, information technology will separate into 2 components, e-ray and o-ray. The e-ray amplitudeA cosθ is parallel to the optic axis and o-ray aamplitudeAsinθ is perpendicular to the optic axis. Every bit shown in Fig. fourteen.32(a), inside the crystal east-ray and o-ray travel in the same direction with dissimilar amplitudes. On emerging from the crystal the rays have a stage difference 'δ' (say), depending on the thickness 'd' of the crystal. Letv be the frequency of lite. The due east-ray and o-ray tin be represented in terms of simple harmonic motions, at correct angles to each other having a phase divergence 'δ'. The e-ray moves faster than o-rays in calcite crystal. Hence, the instantaneous displacements are

x =A cosθ sin (ωt +δ) for east-ray _________ (14.sixty)

andy =A sinθ sinωt for o-ray _________ (14.61)

whereω = 2πν

AllowA cosθ =a andA sinθ =b, then equation (14.60) and equation (14.61) becomes

10 =a sin (ωt +δ) _________ (xiv.62)

y =b sinωt _________ (14.63)

From equation (fourteen.63)

From equation (14.62)

Substituting equation (14.64) in equation (xiv.65)

Squaring both sides

This is the general equation for an ellipse.

Special cases: (1) Suppose the phase differenceδ = 0.

Then sinδ = 0 and cosδ = 1

Equation (14.66) becomes

This is the equation for a straight line. So the calorie-free that comes out of the crystal is airplane polarized.

Case (2): Suppose

due north = 0, 1, 2, three,…

And so, equation (14.66) becomes

Equation (14.68) represents an ellipse. So, the emergent light from the crystal will be elliptically polarized.

Instance (three): Supposeδ = anda =b

And then, from equation (14.68)

x ² +y ² =a ² _________ (14.69)

Equation (14.69) represents a circle. And then, the emergent light from the crystal will exist circularly polarized. Circularly polarized lite can also be produced when the incident plane polarized light makes an bending of 45° with the optic axis. The linear, elliptical and circular polarizations for unlike values ofδ are shown in Fig. 14.33.

Effigy fourteen.33 The different polarizations for differentδ values

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