PHY385 Module 8

                                                                    Student Guide

Concepts of this module

Coherent waves. Phase difference and optical path difference.

Superposition of coherent waves traveling the opposite directions. Standing waves.

Superposition of two incoherent waves. The beats.

Superposition of two coherent waves.

Quiz

Activity 1 - Coherent Waves. Phase difference and optical path difference (OPD).

Coherent waves have equal frequency and wavelength. Their phase difference is constant in time.

Problem 7.6.

Two coherent waves are traveling in parallel directions (see Fig. 1).

Determine the optical path difference for the two waves A and B, both having vacuum wavelength of 500 nm. The glass (n = 1.52) tank is filled with water (n = 1.33). If the waves start out in-phase and all number are exact (the uncertainties are zero), find the phase difference Δα = α₂ - α₁ and the OPD for the two waves at the finish line.

Activity 2 - Superposition of coherent waves traveling in opposite directions. Standing waves.

When a traveling wave is reflected from a stop, the standing wave appears as a sum of the traveling and reflected coherent waves of same amplitude (in absence of absorption). The resultant is a standing wave that does not transfer energy and has zero amplitude at specific locations called the nodes.

Problem 7.13.

Microwaves of frequency 10¹⁰Hz are beamed directly at a metal reflector. Neglecting the refractive index of air, determine the spacing between successive nodes in the resulting standing-wave pattern.

Activity 3 - Superposition of two incoherent waves. The beats.

When a sensor can detect simultaneously two waves with slightly different frequencies, the resultant disturbance in the location of the sensor is called a beat. The beats of two incoherent harmonic waves with same amplitude E₀₁ traveling in the positive x-direction are shown in Fig.2. The result of superposition is a traveling wave E = E₀ (x,t) cos (kx - ωt),

where E₀ = 2E₀₁ cos (k_mx - ω_mt); k = ½ (k₁+ k₂); ω = ½ (ω₁ + ω₂); k_m = ½ (k₁ - k₂); ω_m = ½ (ω₁ - ω₂), (if ω₁ > ω₂, and k₁> k₂). The beat frequency is given by ω_b = ω₁ - ω₂.

Problem 7.15.

Imagine that we strike two tuning forks, one with frequency 0f 340 Hz, the other 342 Hz. What will we hear?

Activity 4 - Superposition of two coherent waves.

EQUIPMENT NEEDED:

- Optics Bench

- Light Source

- Ray Table Base

- Diffraction Scale

- Slit Mask

- Diffraction Plate

- Red Color Filter

- Component Holder

1. Mount equipment on the Optics Bench as shown in Fig.3.

2. Before mounting the Diffraction Plate, center the Slit Mask on the Component Holder on the side facing the Light Source.

3. While looking through the Slit Mask, adjust the position of the Diffraction Scale so you can see the filament of the Light Source through the slot in the Diffraction Scale.

4. Attach the Diffraction Plate to the other side of the Component Holder, as shown. Center pattern D, with the slits vertical, in the aperture of the Slit Mask. Look through the slits. By centering your eye so that you look through both the slits and the window of the Diffraction Plate, you should be able to see clearly both the interference pattern and the illuminated scale on the Diffraction Scale.

In the two-slit experiment, the light wave first falls on a Slit Mask and then on an opaque screen - the Diffraction Plate - with two closely spaced, narrow slits. As Huygens's principle tells us, each slit acts as a new source of light. Since the slits are illuminated by the same wave front, these sources are coherent and in phase. At the location where the wave fronts from the two coherent sources overlap, the superposition of the two waves results in the interference pattern that can be observed either on a screen or directly by eye. NOTE: In this experiment, you look through the narrow slits at the light source, and the interference pattern is formed directly on the retina of your eye (Fig. 4). You then see this interference pattern superimposed on your view of the illuminated diffraction scale.

Fig.4.

The goal of your measurements is to determine the wavelength of light that creates an interference pattern on your retina. The irradiance of the image is proportional to the E².

For the amplitude in a point of observation, the result of superposition of the two coherent waves can be written as

where 2E₀₁E₀₂ cos(α₁ - α₂) is an interference term. For the waves with equal amplitude E₀₁

.

When α₁ - α₂ = 2nπ, n = 0, 1, 2,..., the interference term is at maximum, and you can see the bright line with irradiance that is four times the irradiance of a single source.

When α₁ - α₂ = (2n + 1) π, n = 0, 1, 2,..., the interference term is at minimum, and you see the zero irradiance at the location of superposition of the waves.

The phase difference α₁ - α₂ is related to the OPD as you have found in Activity 1.

Looking through the pair of slits (pattern D) at the Light Source filament, make measurements as explained below with a red filter placed over the Light Source aperture and fill in the table. The value for L of 450 mm is recommended, but can be different in your experiment.

At the zeroth maxima, light rays from slits A and B have traveled the same distance from the slits to your eye, so they are in phase and interfere constructively on your retina. At the first order maxima (to the left of the viewer) light from slit B has traveled one wavelength farther than light from slit A, so the rays are again in phase, and constructive interference occurs at this position as well.

At the nth order maxima, the light from slit B has traveled n wavelengths farther than the light from slit A, so again, constructive interference occurs. In the diagram, the line AC is constructed perpendicular to the line PB. Since the slits are very close together (in the experiment, not the diagram), lines AP and BP are nearly parallel. Therefore, to a very close approximation, AP = CP. This means that, for constructive interference to occur at P, it must be true that BC = nλ.

Notice that θ = arctan X/L, and AB sin (arctan X/L) = nλ.

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Color	n	AB (mm)	X (mm)	L (mm)	Wavelength (nm)	Average (nm)
red	1	0.125		450
	2	0.125		450
	3	0.125		450

This Student Guide was created by Natalia Krasnopolskaia in November 2014.