9.3 Interference

Students in a laboratory have created the following set-up. Parallel rays of monochromatic light from two adjacent slits A and B of wavelength $\lambda$ are incident normally on a diffraction grating with a slit separation $\mathrm{d.}$

Use the diagram to derive the equation nλ = dsinθ where θ is the angle of diffraction of a maxima order n visible on a screen.

The monochromatic light in the set-up in part (a) has a wavelength of 545 nm. The graph shows the variation of sinθ with the order n of the maximum. The central order corresponds to n = 0.

Determine a mean value for the number of slits per mm of the grating.

The grating is 40 mm wide.

Determine the number of slits required to obtain a maximum of six bright fringes on the screen.

The students claim that they observed the following diffraction pattern on the screen for the grating from part (c).

State two reasons why the interference pattern obtained cannot be correct. 

An internet company is looking to improve the amount of light transmitted through its optical fibres. The fibres are made of glass and have a refractive index n_g.

The proposed solution has been to spread a thin film of oil on the inside surface with a refractive index n_o. The air inside the fibre has a refractive index n_a and the air outside has a refractive index n.

n < n_a < n_o < n_g 

Complete the ray diagram to show the path of the incident light ray along the optical fibre.

You do not need to indicate any angles of refraction.

Outline the conditions for constructive interference to occur as two light rays travel down in an optical fibre with oil on the inside.

You may include a diagram in your answer.

The oil has a thickness of 100 nm and a refractive index of 1.4. 

Determine the longest possible wavelength of light that can be incident on the oil to obtain constructive interference.

Analyse whether the oil improves the amount of light transmitted through the optical fibres. 

The diagram below shows an arrangement for observing the interference pattern produced by laser light passing through two narrow slits S₁ and S₂.

The distance S₁S₂ is d, and the distance between the double slit and the screen is D where D ≫ d, so angles θ and ϕ are small. M is the midpoint of S₁S₂ and it is observed that there is a bright fringe at point A on the screen, a distance f_n from point O on the screen. Light from S₁ travels a distance S₂Y further to point A than light from S₁.

The wavelength of light from the laser is 650 nm and the angular separation of the bright fringes on the screen is 5.00 × 10⁻⁴ rad. Calculate the distance between the two slits.

A bright fringe is observed at A. 

Deduce expressions for the following angles in the double-slit arrangement shown in part a: 

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The separation of the slits S₁ and S₂ is 1.30 mm. The distance MO is 1.40 m. The distance f_n is the distance of the ninth bright fringe from O and the angle θ is 3.70 × 10⁻³ radians. 

Calculate the wavelength of the laser light. 

Monochromatic light is incident normally on four, thin, parallel, rectangular slits. 

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The graph shows the variation with diffraction angle θ of the intensity of light I on a distant screen. 

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I₀ is the intensity of the light at the middle of the screen from one slit. 

State the value of the light intensity in terms of I₀  when θ = 0 and explain where this value comes from.

The width of each slit is 2.0 µm. 

Use the graph to estimate the wavelength of the light.

Use the graph to calculate the number of lines per meter on the diffraction grating. 

The four slits are now changed for a grating where the number of slits becomes very large. The separation of the slits and their width stays the same.

State two changes to the graph that will appear as a result of this modification.

Monochromatic light is incident on a thin film of transparent plastic as shown below. 

The plastic film is in the air. 

Light is partially reflected at both surfaces X and Y on the film. 

State and explain the reasons for the phase change that occurs when light is reflected from:

The red light incident on the transparent plastic has a wavelength of 630 nm. The refractive index of the plastic is 1.50. 

Calculate the minimum thickness of the plastic for the light reflected from surface X and surface Y to undergo constructive interference. 

In the second investigation, a thin film of colourless oil floats on water as shown in the diagram below. The refractive index of the oil is 1.47 and the water is 1.52.  The same red light is now incident on the oil.  

Complete the diagram to show the two light rays reflected from the two surfaces of the oil and label them P and Q.

The observer notices that the red light reflected from the oil is now darker than that reflected from the transparent plastic. 

Calculate how many times thicker the thinnest film of oil is compared to the thinnest film of transparent plastic.

Monochromatic light is incident on a double-slit diffraction grating. After passing through the slits the light is brought to a focus on a screen. The intensity distribution of the light on the screen is shown in the diagram below. 

The double-slit diffraction grating is now changed to a grating with many narrower slits, the same widths as the slits above. 

Sketch the new intensity pattern for the light between points C and D on the screen. 

The wavelength of the monochromatic light incident on the diffraction grating is 550 nm. The slit spacing of the diffraction grating is 1.34 × 10⁻⁶ m.

Calculate the angle between the two second-order maxima. 

Calculate the total number of orders of diffracted light that can be observed on the screen. 

Two sources of light now replace the light incident on the diffraction grating. One is the same as the wavelength of the previous source and the other has a slightly longer wavelength. 

Compare and contrast the new intensity pattern with the original. Comment on the intensity of the central maxima and the width of all maxima.