Polarization of light - II: Assignment | Oscillations, Waves & Optics - Physics PDF Download

Q.1. A HWP (half-wave plate) is introduced between two crossed Polaroids P₁ and P₂. The optics axis makes an angle 15^o with the pass axis of P₁ as shown in figure (i) and (ii)  given below. If an unpolarized beam of intensity I₀ is normally incident on P₁ and if I₁,I₂ and I₃ are the intensities after P₁ , after HWP and after P₂ respectively then calculate I₁/I₀/I₂/I₀ and I₃ / I₀ .

(a)

(b)

(c)

where θ is the angle between optic axis and pass axis P₁ and δ is the phase difference between O - ray and e - ray at  the exit of Half wave plate.

For half wave plate ( HWP ) ,δ =π

∴

Given θ = 15⁰

∴

⇒

Q.2. A quarter - wave plate is rotated between two crossed polaroids. If a polarized beam is incident on the first Polaroid, discuss the variation of intensity of the emergent beam as the quarter - wave plate is rotated. What will happen if we have a half - wave instead of a quarter - wave plate?

The intensity of light at the exit of Polaroid P₂ is

 

(a) For QWP : δ =π / 2

∴

(b) For HWP :   δ =π

∴

Q.3. Consider a calcite HWP whose optic axis is along the y - axis, obtain the output state of polarization when the incident beam is

(a) x - polarized

(b) y - polarized

(c) Left Circularly Polarized (LCP)

(d) Linearly Polarized with its E making an angle of 45^o with the y - axis

(e) Linearly Polarized with its E making an angle of 60⁰ with the x - axis

(f) Left Elliptically Polarized (LEP) with its E given by

(a)

(b)

(c)

(d)

(e)

(f)

Q.4. Two prisms of calcite ( n_o > n_e ) are cemented together as shown in the figure given below so as to form a cube. Lines and dots show the direction of the optic axis. A beam of unpolarized light is incident normally from region I. Assume the angle of the prism to be 12^o. Determine the path of rays in region II, III and IV indicating the direction of vibrations (i.e. the direction of D).

It is example of Rochon Prism. In the first prism both the beams will see the same refractive index n . In second crystal, the o - ray will see the same refractive index and go undeviated. The e - ray will see refractive index n_e and will bend away from the normal. The angle of refraction will be determined from the following equation:

 

Thus, sinr =

sin r = 0.232 ⇒  r ≅13.4⁰

Therefore, the angle of incidence at the second surface will be 13.4⁰ -12⁰= 1.4⁰

The emerging angle will be sinθ = n_esin(1.4) ⇒θ  ≅ 2.1⁰

Q.5. For calcite the values of n₀ and n_e for λ₀ = 4046A`⁰ are 1.68134 and 1.49694 respectively; corresponding to λ₀ = 7065A⁰ the values are 1.65207 and 1.48359, respectively. We have a calcite quarter – wave plate corresponding to λ₀ = 4046A⁰. A left – circularly polarized beam of λ₀ = 7065A⁰ is incident on this plate. Obtain the state of polarization of the emergent beam.

Thickness of the calcite plate is 

d =  = 5485 x 10^-7mm

For  λ₀ = 7065A⁰

δ =  =

Equation of left circularly polarized beams

After passing through calcite crystal of thickness d , additional phase difference of π/4 is 

introduced between E_x and E_y

∴

It represents the left elliptically polarized beam.

Q.6. (a) Consider two crossed polaroids placed in the path of placed in the path of an unpolarized beam of intensity I₀ (see figure). If we place a third polariod in between the two then, in general, some light will be transmitted through. Explain this phenomenon.

(b) Assuming the pass axis of the third polaroid to be at 0 45 to the pass axis of either of the polaroids, calculate the intensity of the transmitted beam. Assume that the polaroids are perfect.

(a)

Let angle between pass axis P₁ and P₂ is θ the intensity at the exit of P₂ is

Since angle between P₁ and P₃ is 90⁰ Therefore angle between P₂ and P₃ is 90⁰ -θ

∴ Intensity at the exit of P₃ is

 =  =

(b) θ = 45⁰

∴  

Q.7. In question 3, if the optic axis of the quarter-wave plate makes an angle of 45⁰ with the pass axis of either Polaroid, show that only a quarter of the incident intensity will be transmitted. If the quarter – wave plate is replaced by a half - wave plate is replaced by a half - wave plate, show that half of the incident intensity will be transmitted through.

(a) For QWP : θ =π /2

∴

For θ = 45⁰ :

(b) for HWP : δ =π

∴

For θ = 45⁰ :

Q.8. A λ/6 plate is introduced in between the two crossed polarization in such a way that the optic axis of the λ/6 plate makes an angle of 45⁰ with the pass axis of the first polarizer (figure given below). Consider an unpolarized beam of intensity I₀ to be incident normally on the polarizer. Assume the optic axis to be along the z - axis and the propagation along the x - axis. Write the y and z components of the electric fields (and the corresponding total intensities) after passing through (i) P₁ (ii) λ/6 plate and (iii) P₂ .

(i) Intensity after P₁- is

(ii) Intensity after P₂ is

(iii) Intensity after P₃ is

where θ is the angle between optic axis and pass axis of P₁ and δ is the phase difference between O - ray and e - ray introduced by λ/6- plate. 

Given θ =

∴

Q.9. Consider a Wollaston prism consisting of two similar prisms of calcite ( n_o = 1.66 and n_e = 1.49) with angle of prism 25⁰. Calculate the angular divergence of the two emerging beams.

For z - polarized beam: The beam will propagate as an o - ray in first prism with refractive index n_o = 1.66 . When this beam enters the second prism it will become an e - ray with n_e = 1.49

∴  ≌ 25⁰

Thus the angle of incident at the second surface will be 

i₁ = 28- 25= 3⁰

The output angle θ₁ will be given by

For y - polarized beam: The beam will propagate as e - ray in the first prism and as an o - ray in the second prism. 

The angle of refraction will now be given by

 =  22.3⁰
Thus, the angle of incident at the second surface will be  

i₂ = 25⁰ - 22.3⁰ = 2.7⁰

The output angle θ₂ will be given by

∴ Angular divergence is θ = θ₁ + θ₂≌9⁰