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PH113 / Assignment - Module 1: Physical Optics

Here, we present some of the benchmark question, mostly numerical, from Module I to be
discussed in the tutorials. It is expected that the students will solve these questions on their
own, or in groups before the tutorial(s). At the same time, questions similar to the ones
contained herein are likely to be asked in your exams. However, note that (i) same questions
should NOT be expected in the exams, (ii) numerical questions beyond these could also be
asked in the exams. In other words, this document is NOT a question bank of any kind.
Therefore, the students should aim to understand the underlying concepts well.

1. Malus Law.

(a) Consider a geometry of sequence of two polarizers rotated at a relative angle of 45◦
from each other. What would be resulting amplitude and intensity if the incident
unpolarized (light) beam had the intensity I0 ?

(b) Similar to above, consider a sequence of three polarizers at relative angles of 20◦ to
each other. Also, the first polarizer is aligned along the y-axis. Compute the intensities
Ii (i = 1, 2, 3) after each polarizer if the intensity of the y-polarized source is given to
be I0 = 1000 W/m2 .

(c) What happens to the final intensity if the second polarizer is removed?

(d) [!]It is rather straight-forward to generalize the above situation to n polarizers separated
by angle θ (fixed!). In principle, it would be interesting to find out when the resulting
intensity will be ≲ 0.10I0 .

Comment: At-home exercise: Find Polaroid sunglasses and rotate one while holding
the other still and look at different surfaces and objects. Explain your observations.
What is the difference in angle from when you see a maximum intensity to when you
see a minimum intensity? Find a reflective glass surface and do the same. At what
angle does the glass need to be oriented to give minimum glare?

2. Brewster angle

On a calm surface of a pond or a lake (interface between air and water), for which position
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of the sun would the reflected light from the interface be completely linearly polarized. The
refractive indexes for air and water should be taken as nair = 1 and nwater = 1.33.

Comment: This situation is, in fact, realized at the beaches or lakes.

3. A plate of flint glass (refractive index 1.67) is immersed in water (refractive index 1.33).
Calculate the Brewster angle for internal as well as external reflection at an interface.

4. [!] At a particular beach on a particular day near sundown the horizontal component of
the electric field vector is 2.3 times the vertical component. A standing sunbather puts
on polarizing sunglasses; the glasses suppress the horizontal component. (a) What fraction
of light intensity received before the glasses were put on now reaches the eyes? (b) The
sunbather, still wearing the glasses lies on his side. What fraction of the light energy received
before the glasses were put on reaches the eyes now?

5. Double refraction.

A thin quartz crystal of thickness t has been cut so that the optical axis is ∥ to the surface
of the crystal. For sodium light (λ = 589 nm), the refractive index of the crystal is 1.55 for
e-ray and 1.54 for o-ray. If two beams of light polarized ∥ and ⊥ O.A., respectively, start
through the crystal in-phase, how thick must the crystal be if the emerging rays have a phase
difference of π/2.

6. Calculate the thickness of a doubly refracting crystal required to introduce a path difference
of λ/2 between the e-ray and o-ray when λ = 60000 Å, µo = 1.65 and µe = 1.54.

7. Plane waves and their superposition

(a) Plot y = y0 cos(ωt − kx + ϕ) for (i) ω = 1 s−1 , (ii) ω = 0.5 s−1 and ϕ = 0,±π/4, ±π/2,
±π, ±2π. What are the units of ϕ, ω and k?

Comment: This should be useful.

(b) Consider superposition of two plane waves of equal amplitude and in-phase (ϕ = 0),
polarized along x̂ and ŷ, respectively. What would be resultant? (Find the amplitude
and polarization of the resultant wave).
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(c) In the above, consider the phase-difference between the two waves to be fixed at ϕ = π.
How does this affect the resultant wave?

(d) Show that the the amplitude of the wave obtained by superposition of two waves of
equal amplitude and a phase difference of ϕ = π/2 is equal to the amplitude of the
initial waves. What does the wavefront of the superposed wave look like?

What happens for ϕ = −π/2?

Comment: The phase difference of π/2 leads to circularly polarized light. Depending on
the sign of the phase difference, one obtains left or right circularly polarized light. The
nomenclature for left/right is somewhat subjective in literature. However, an accepted
practice is to look towards the direction of propagation.

8. The magnetic field equations for an electromagnetic wave in free space are

Bx = Bsin(ky + ωt)

By = 0

Bz = 0 .

(a) What is the direction of propagation? (b) Write the electric field equations. (c) Is the
wave polarized? If so, in which direction.

9. Interference

Prove the cosine law: the path difference between the interfering beams in a (parallel ) thin
film geometry at oblique incidence is 2µt cos θr , where µ is the refractive index of the denser
medium, t is the thickness of the medium, and θr is the angle of refraction with respect to
normal.

10. [!] A thin glass of refractive index 1.5 is placed in the path and the central bright band is now
shifted to the position of the third bright band. The fringes are produced by monochromatic
light of wavelength 5450 Å. Calculate the thickness of the glass. You may assume any suitable
geometry for interference.

Comment: (Hint) Compute the optical path difference due to (additional) glass in the path
of the second interfering wave and equate it to twice/thrice the fringe width.
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11. A thin film of thickness 4 ×10−5 cm is illuminated by white light normal to its surface. Its
refractive index is 1.5. Of what color will the thin film appear in reflected light?

12. White light is reflected normally from a uniform oil film (n = 1.33). An interference max-
imum for 6000 Å and a minimum for 4500 Å with no minimum in between, are observed.
Calculate the thickness of the film.

13. [!] Consider a light incident on parallel plate(s) of refractive index n2 = 1.414 (n1 = 1 for
air) at an angle on 45◦ . Obtain the possible values of wavelength which will produce bright
fringes if the thickness of the parallel plates is t.

14. [!] Prove that two linearly polarized light waves of equal amplitude, with their planes of
vibration at right angles to each other, cannot produce interference effects. (Hint: Prove
that the intensity of the resultant light wave, averaged over one or more cycles of oscillation,
is the same no matter what phase difference exists between two waves.)

15. Interference from wedge-shaped surfaces

Light of wavelength λ = 6300 Å falls normally on a thin wedge-shaped film (n = 1.5). There
are ten bright and nine dark fringes over the length of the film. By how much does the film
thickness change over this length?

16. Two glass plates 12 cm long touch at one end, and are separated by a wire 0.048 mm in
diameter at the other. How many bright fringes will be observed over the 12 cm distance in
the light (λ = 6800 Å ) reflected normally from the plates? (Take µ = 1).

17. Newton’s rings

(a) What is an efficient (and presumably error-free) way to obtain λ?

(b) Show that in the Newton’s rings arrangement, if the plano-convex lens is slowly lifted
vertically upwards, the fringes will move inwards.

(c) Newton’s rings are formed in reflected light of wavelength 5895 Å with a liquid between
the plane and curved surfaces. The diameter of the 5th ring is 0.3 cm and the radius of
curvature of the curved surface is 100 cm. Calculate the refractive index of the liquid,
when the 5th ring is i) bright, ii) dark.
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Comment: Explicitly derive the conditions for observing a bright spot at the center.

18. Solution to the transcendental equation tan(x) = x.

19. Diffraction from a single slit.

Sound wave has a speed of 330 m/s, and suppose it’s frequency is 100 Hz. What would be
the typical aperture size to experience appreciable diffraction?

20. Diffraction from 2-slit arrangement.

Explicitly derive the conditions for maxima/minima. Note and plot the intensity profile.

21. Computational/Simulation exercise(s).

Plot the resulting intensity patterns from (i) a single slit diffraction; (ii) double slit diffraction.

Solution. Check out these links:

• Single slit: Geogebra - based on the formula

• Young’s double slit: Geogebra, Javalab or, oPhysics

Comment: Study the dependence on the slit width, etc.

22. Light matter interaction. To provide a physical intuition of how light interacts with mat-
ter to produce polarization and related effects, you may have heard about the fact that oscil-
lating EM waves induce oscillations in electric dipoles. Here is a nice illustration/animation
of the same even if the context is limited.