Notes-Part-1-Class-12-Physics-Chapter-7-Wave Optics-MSBSHSE

Drawbacks of Newton's corpuscular theory of light :

• The theory predicted that the speed of light in a denser medium should be greater than that in air. This was disproved when experiment showed that the speed of light in water is less than that in air (carried out in 1850 by French physicist Jean Bernard Le'on Foucault).
• The theory could not satisfactorily explain the phenomenon of polarization and the simultaneousness of reflection and refraction.
• The corpuscular theory failed to explain the phenomena of diffraction and interference.
• There was no basis for the hypothesis that the constituent colours of white light are due to different sized corpuscles.

Merits and Demerits of theory :

Maxwell's concept of light :

In 1865, James Clerk Maxwell developed a mathematical theory on the intimate relationship between electricity and magnetism. His theory predicted :

• Light to be a high-frequency transverse electromagnetic wave in ether.
• Electric and magnetic fields in the wave vary periodically in space and time at right angles to each other and to the direction of propagation of the wave.

The speed of the electromagnetic waves in a medium, as calculated on the basis of Maxwell's theory, was experimentally found to be equal to the measured speed of light in that medium.

Maxwell's electromagnetic theory of light, with addition by others till 1896, could account for all the known phenomena regarding the propagation (or transmission) of light through space and through matter.

Characteristics of the electromagnetic waves :

• The electromagnetic waves are transverse in nature as they propagate by oscillating electric and magnetic fields that are perpendicular to each other and also perpendicular to the direction of propagation of the wave.
• These waves do not require any material medium for their propagation, i.e., they can travel even through vacuum.
• The wavelength of the electromagnetic waves ranges from very small ( < 1 fm) to very large (> 1 km). The waves are classified in the order of increasing wavelength as y-rays, X rays, ultraviolet, visible, infrared, microwave and radio waves.
• In vacuum, the speed of electromagnetic waves does not depend on the frequency of the wave. But, in a material medium, it depends on the frequency.
• For a given frequency, the speed is different in different mediums.

[1 fm (femtometre) = 10^-15 m]

Wavefront related to rays of light :

Consider a point source of light O in a homogeneous isotropic medium in which the speed of light is v. The source emits light in all directions. In time t, the disturbance (light energy) from the source, covers a distance vt in all directions, i.e, it reaches out to all points which are at a distance vt from the point source. The locus of these points which are in the same phase is the surface of a sphere with the centre O and radius vt. It is a spherical wavefront.

Click here to View Figure-1 

In a given medium, a set of straight lines can be drawn which are perpendicular to the wavefront.

According to Huygens, these straight lines are the rays of light. Thus, rays are always normal to the wavefront. In the case of a spherical wavefront, the rays are radial.

Wavenormal and Ray of light :

Huygens’ construction of a plane wavefront : A plane wavefront may be treated as a part of a spherical or cylindrical wave at a very great distance from a point source or an extended source, such that the wavefront has a negligible curvature.

Let us first consider a plane wavefront AB (corresponding to parallel rays), at time t = 0 cross section of which is shown in above Fig..

Click here to View Figure-5 

Let P, Q, R, S, ..., be points on a plane wavefront in a homogeneous isotropic medium in which the speed of light, taken to be monochromatic, is v (Fig.).

In a time, t= T, secondary wavelets with points P, Q, R, S,..., as secondary sources travel a distance vT. To find the position of the wavefront after a time t = T, we draw spheres of radii vT with P, Q, R, S,..., as centres. The envelope or the surface of tangency to these spheres is a plane P’Q’R’S’. This plane, the new wavefront A’B’, is at a perpendicular distance vT from the original wavefront in the direction of propagation of the wave. Thus, in an isotropic medium, plane wavefronts are propagated as planes.

Huygens’ construction of a spherical wavefront :

Consider a point source of monochromatic light S in a homogeneous isotropic medium. The light waves travel with the same speed v in all directions. After time t, the wave will reach all the points which are at a distance vT from S. This is spherical wavefront XY. Let, A, B, C, ...,, be points on this wavefront.

Click here to View Figure-6 

To find the new wavefront after time T, we draw spheres of radius vT with A, B, C,… as centres. The envelope or the surface of tangency of these spheres is the surface A’B’C’. This is the new spherical wavefront X’Y’. Thus, in an isotropic medium, spherical wavefronts are propagated as concentric spheres.

Laws of reflection of light using Huygens’ principle :

Consider a plane wavefront AB of monochromatic light incident obliquely at an angle i on a plane mirror MN. The wavefront AB touches the reflecting surface MN at A at time t = 0. Let v be the speed of light in the medium.

When wavefront AB is incident on the mirror, at first, point A becomes a secondary source and emits secondary waves in the same medium. If T is the time taken by the incident wavefront to travel from B to C, then BC = vT. During this time, the secondary wave originating at A covers the same distance, so that the secondary spherical wavelet has a radius vT at time T.

Click here to View Figure-7 

To construct the reflected wavefront, a hemisphere of radius vT is drawn from point A Draw a tangent EC to the secondary wavelet.

As all points on EC are in the same phase of wave motion, EC represents the reflected wavefront.

The arrow AE shows the direction of propagation of the reflected wave.

AP is the normal to MN at A,

∠ RAP = i = angle of incidence and

∠ PAE = r = angle of reflection

In Δ ABC and Δ AEC,

AC is common,

AE =BC and ∠ ABC = ∠ AEC = 90°

 Δ ABC and Δ AEC are congruent.

 ∠ ACE = ∠ BAC = i  …….(1)

Also, as AE is perpendicular to CE and AP is perpendicular to AC,

∠ ACE = ∠ PAE =r ...(2)

 From Eqs (1) and (2),

i = r

Thus, the angle of incidence is equal to the angle of reflection. This is the first law of reflection. Also, it can be seen from the figure that the incident ray and reflected ray lie on the opposite sides of the normal to the reflecting surface at the point of incidence and all of them lie in the same plane. This is the second law of reflection. Thus, the laws of reflection of light can be deduced by Huygens’ construction of a plane wavefront.

CE = AB shows that the size of the image equals that of the object. On reflection, A is seen at E and B is seen at C. This is lateral inversion, i.e., on reflection, the right side of an object becomes the left side, and the left side of the object becomes the right side.

Laws of refraction of light using Huygens’ principle

OR

Snell's law of refraction of light on the basis of wave theory :

Consider a plane wavefront AB of monochromatic light propagating in the direction A’A incident obliquely at an angle i on a plane refracting surface MN. This plane refracting surface MN separates two uniform and optically transparent mediums.

Let v₁ and v₂ be the speeds of light in medium 1 (say, a rarer medium) and medium 2 (a denser medium) respectively.

Click here to View Figure-8 

When the wavefront reaches MN at point A at t = 0, A becomes a secondary source and emits secondary waves in the second medium, while ray B’B reaches the surface MN at C at time t = T. Thus, BC = v₁T. During the time T, the secondary wavelet originating at A covers a distance AE in the denser medium with radius v₂T.

As all the points on CE are in the same phase of wave motion, CE represents the refracted wavefront in the denser medium. CE is the tangent to the secondary wavelet starting from A. It is also a common tangent to all the secondary wavelets emitted by points between A and C. PP’ is the normal to the boundary at A.

∠ A’AP = ∠ BAC = the angle of incidence (i) and

∠ P’AE = ∠ ACE = the angle of refraction (r).

From Δ ABC and Δ AEC,

sin i = \(\frac{BC}{AC}\)  and sin r = \(\frac{AE}{AC}\) 

∴ \(\frac{sin\,i}{sin\,r}=\frac{BC/AC}{AE/AC}=\frac{BC}{AE}=\frac{v_1T}{v_2T}=\frac{v_1}{v_2}\)

By definition, the refractive index of medium 2 with respect to medium 1,

₁n₂ = \(\frac{n_2}{n_1}=\frac{v_1}{v_2}\)

∴ \(\frac{n_2}{n_1}=\frac{sin\,i}{sin\,r}\)

n₁ sin i  = n₂ sin r    ….. (1)

Here, n₁ and n₂ are the absolute refractive indices of medium 1 and medium 2 respectively. Eq. (1) is Snell's law of refraction. Also, it can be seen from the figure, that the incident ray and the refracted ray lie on the opposite sides of the normal and all three of them lie in the same plane.

Thus, the laws of refraction of light can be deduced by Huygens’ construction of a plane wavefront.

If v₁ > v₂, i.e. n₁ < n₂, then r < i (bending of the refracted ray towards the normal).

Dependence of Wavelength on the Refractive Index of the Medium:

Consider monochromatic light incident normally on a boundary between a rarer medium and a denser medium as shown in Fig. (a).

The boundary between the two surfaces is shown by PQ.

The three successive wavefronts AB, CD and EF are separated by a distance λ₁, which is the wavelength of light in the first medium.

After refraction, the three wavefronts are indicated by A′B′, C′D′ and E′F′.

Click here to View Figure-9 

Assuming the second medium to be denser, the speed of light will be smaller in that medium and hence, the wavefronts will move slower and will be able to cover less distance than that covered in the same time in the first medium.

They will therefore be more closely spaced than in the first medium. The distance between any two wavefronts is λ₂, equal to the wavelength of light in the second medium. Thus, λ₂ will be smaller than the wavelength in the first medium.

We can easily find the relation between λ₁ and λ₂ as follows.

Let the wavefront AB reach the boundary surface PQ at time t = 0 and the next wavefront CD which is at a distance of λ₁ from AB, reach PQ at time t = T. As the speed of the wave is v₁ in medium 1 and T is the time period in which the distance λ is covered by the wavefront , we can write

T = λ₁/v₁   ….(1)

In medium 2, the distance travelled by the wavefront in time T will be λ₂. The relation between these two quantities will be given by

T  = λ₂/v₂  …….(2)

Eq. (1) and Eq. (2) show that the velocity in a medium is proportional to the wavelength in that medium and give

λ₂ = λ₁ v₂/ v₁ =  λ₁ n₁/n₂  ……..(3)

If medium 1 is vacuum where the wavelength of light is λ₀ and n is the refractive index of medium 2, then the wavelength of light in medium 2, λ, can be written as

λ = λ₀ v₂/c = λ₀/n …. (4) (… as v₁ = c)

The ratio of the frequencies ν₁ and ν₂, of the wave in the two media can be written, using Eq. (3) as,

ν₁/ν₂ = (v₁/λ₁)/(v₂/λ₂) = 1 …..(5)

This demonstrates that the frequency of a wave remains unchanged while going from one medium to another.

Thus, ν_o = ν₁ = ν₂, where ν_o is the frequency of light in vacuum.

Similar analysis goes through if the wave is incident at an angle as shown in Fig. (b)

Click here to View Figure-10 

We have

λ₂ = λ₁ n₁/n₂  and ν₁ = ν₂

Distinguish between polarized and unpolarized light in a ray diagram :

Linearly polarized light is represented in a ray diagram by double-headed arrows or short lines drawn perpendicular to the direction of propagation of light, as shown in Fig. (a).

Click here to View Figure-11 (a) and (b) 

An unpolarized light beam is represented by both dots and arrows, as shown in Fig. (b). The dots are the ‘end views’ of arrows that are oriented normal to the plane of the diagram.

Note : The length of an arrow or line represents the amplitude of the electric field (E ) in the plane of the diagram and the direction indicates the polarization axis of the beam, i.e., the direction of vibration of . According to Jean Biot (1774-1862), French physicist, unpolarized light may be considered as a superposition of many linearly polarized waves, with random orientations in planes perpendicular to the direction of propagation.

In an analytical treatment, the electric vectors may be resolved along two mutually perpendicular directions so that the multiplicity of vectors in below Fig. (a) can be replaced by the two mutually perpendicular vectors as in below Fig. (b), both perpendicular to the direction of propagation of light. Sir David Brewster (1781-1868), British physicist, conceived the ordinary unpolarized light to consist of two perpendicular, polarized components of equal intensity.

Click here to View Figure-12 (a) and (b) 

Polarization and Malus’ law : Malus’ law gives the intensity of a linearly polarized wave after it passes through a polarizer.

According to the electromagnetic theory of light, a light wave consists of electric and magnetic fields vibrating at right angles to each other and to the direction of propagation of the wave. If the vibrations of \(\vec{E}\)   in a light wave are in all directions perpendicular to the direction of propagation of light, the wave is said to be unpolarized.

If the vibrations of the electric field \(\vec{E}\) in a light wave are confined to a single plane containing the direction of propagation of the wave so that its electric field is restricted along one particular direction at right angles to the direction of propagation of the wave, the wave is said to be plane-polarized or linearly polarized.

This phenomenon of restricting the vibrations of light, i.e., of the electric field vector in a particular direction, which is perpendicular to the direction of the propagation of the wave is called polarization of light.

Click here to View Figure-14 

 Consider an unpolarized light wave travelling along the x-direction. Let c, n and λ be the speed, frequency and wavelength, respectively, of the wave. The magnitude of its electric field (\(\vec{E}\) ) is,

E = E₀ sin (kx − ωt), where E₀ = E_max = amplitude of the wave, ω = 2pn = angular frequency of the wave and k = 2p/λ = magnitude of the wave vector or propagation vector.

The intensity of the wave is proportional to |E₀|².

The direction of the electric field can be anywhere in the y-z plane. This wave is passed through two identical polarizers as shown in below Fig.

When a wave with its electric field inclined at an angle Φ to the axis of the first polarizer is passed through the polarizer, the component E₀ cos Φ  will pass through it. The other component E₀ sin Φ which is perpendicular to it will be blocked.

Click here to View Figure-15 

Now, after passing through this polarizer, the intensity of this wave will be proportional to the square of its amplitude, i.e., proportional to |E₀ cos Φ|².

The intensity of the plane-polarized wave emerging from the first polarizer can be obtained by averaging |E₀ cos Φ|² over all values of Φ between 0 and 180°. The intensity of the wave will be proportional to ½|E₀|²  as the average value of cos²Φ  over this range is  Thus the intensity of an unpolarized wave reduces by half after passing through a polarizer.

When the plane-polarized wave emerges from the first polarizer, let us assume that its electric field ( ) is along the y-direction. See Fig . Thus, this electric field is,

\(\vec{E}\) =\(\hat{j}\) E₁₀ sin (kx—ωt)  …..(1)

where, E₁₀ is the amplitude of this polarized wave.

The intensity of the polarized wave,

I₁ ∝ lE₁₀l²  …….(2)

Now this wave passes through the second polarizer whose polarization axis (transmission axis) makes an angle θ with the y-direction. This allows only the component E₁₀ cos θ to pass through it.

Thus, the amplitude of the wave which passes through the second polarizer is

E₂₀ = E₁₀ cos θ and its intensity,

I₂ ∝ |E₂₀|²

∴ I₂ ∝ |E₂₀|² cos²θ

∴ I₂ = I₁ cos²θ   ….(3)

Thus, when plane-polarized light of intensity I₁ is incident on the second identical polarizer, the intensity of light transmitted by the second polarizer varies as cos²θ, i.e., I₂ = I₁ cos²θ, where θ is the angle between the transmission axes of the two polarizers. This is known as Malus’ law.

Use of a pair of polarizers to vary the intensity of light :

Consider an unpolarized light beam incident perpendicularly on the first polarizing sheet, called the polarizer, whose transmission axis is vertical, say.

Click here to View Figure-16 

The light emerging from this sheet is polarized vertically, and the transmitted electric field is \(\vec{E_0}\)  The polarized light beam then passes through a second polarizing sheet, called the analyser, which is placed parallel to the polarizer with its transmission axis at an angle θ  to the transmission axis of the polarizer.

The component of \(\vec{E_0}\) which is perpendicular to the axis of the analyser is completely absorbed, and the component parallel to that axis is E₀ cos θ

Since intensity varies as the square of the amplitude, the transmitted intensity varies as

I = I₀ cos²θ

where I₀ is the incident light intensity on the analyser. This expression (known as Malus’s law) shows that the transmitted intensity is maximum when the transmission axes are parallel and zero when the transmission axes are perpendicular to each other.

Polarization by Reflection: Brewster’s Law:

Consider a ray of unpolarized monochromatic light incident at an angle θ_B on a boundary between two transparent media as shown in Fig. Medium 1 is a rarer medium with refractive index n₁ and medium 2 is a denser medium with refractive index n₂. Part of incident light gets refracted and the rest gets reflected. The degree of polarization of the reflected ray varies with the angle of incidence.

The electric field of the incident wave is in the plane perpendicular to the direction of propagation of incident light. This electric field can be resolved into a component parallel to the plane of the paper, shown by double arrows, and a component perpendicular to the plane of the paper shown by dots, both having equal magnitude. Generally, the reflected and refracted rays are partially polarized, i.e., the two components do not have equal magnitude.

Click here to View Figure-17 

In 1812, Sir David Brewster discovered that for a particular angle of incidence. θ_B, the reflected wave is completely plane-polarized with its electric field perpendicular to the plane of the paper while the refracted wave is partially polarized. This particular angle of incidence (θ_B) is called the Brewster angle.

For this angle of incidence, the refracted and reflected rays are perpendicular to each other.

For angle of refraction θ_r,

θ_B + θ_r = 90° ...(1)

From Snell's law of refraction,

n₁ sin θ_B = n₂ sin θ_r   ….(2)

From Eqs. (1) and (2), we have,

n₁ sin θ_B = n₂ sin(90°— θ_B) = n₂ cos θ_B  

∴ θ_B = tan^-1\(\frac{n_2}{n_1}\)

This is called Brewster's law.

∴ Brewster's law : The tangent of the polarizing angle is equal to the refractive index of the reflecting medium with respect to the surrounding (₁n₂).

If θ_B is the polarizing angle,

tan θ_B  = ₁n₂ = \(\frac{n_2}{n_1}\)

Here n₁ is the absolute refractive index of the surrounding and n2 is that of the reflecting medium.

The angle θ_B is called the Brewster angle.

Uses of Polaroid :

• A Polaroid lens filter makes use of polarization by reflection. This filter is used in photography to reduce or eliminate glare from reflective nonmetallic surfaces like glass, rock faces, water and foliage. It can also deepen the colour of the skies.
• Polaroid sunglasses reduce the transmitted intensity by a factor of one half and also reduce or eliminate glare from nonmetallic surfaces such as asphalt roadways and snow fields.
• Polaroid filters are used in liquid crystal display (LCD) screens.
• Polaroids are used to produce and show 3-D movies to give the viewer a perception of depth.

Polarization by Scattering:

When Sunlight strikes air molecules or dust particles in the atmosphere, it changes its direction. This is called scattering.

The scattered light observed in a direction perpendicular to the direction of incidence, is plane polarized. This phenomenon is called polarization by scattering.

Click here to View Figure-18 

As shown in figure, a beam of an unpolarized light is incident along the Z-axis on a molecule. Light waves being transverse in nature, all the possible directions of vibration of electric field vector in the unpolarized light are confined to the XY plane. The light incident on the molecule is scattered by the electromagnetic field of the molecule.

When observed along the X-axis, only the vibrations of electric field vector which are parallel to Y-axis can be seen. In the similar manner, when observed along the Y-axis, only the vibrations of electric field vector which are parallel to the X-axis can be seen. Thus, the light scattered in a direction perpendicular to the incident light is plane polarized.