Diffraction and Interference Study Guide (page 3)
In this lesson, we will address the second approach to the concept of light: waves. We will study the superposition principle, diffraction, Huygens' principle, constructive and destructive interference, diffraction gratings, and polarization.
We have studied harmonic motion and the time dependence of perturbation. We will now deepen our understanding of waves by studying their interaction. Because each wave is a time and spatial variation of a perturbation and because the appearance of the wave might be different depending on the relative distance between the point of study and the source, we are looking for an expression that connects all these elements together: time, displacement, and propagation. The expression is
y(x, t) = A · sin (k · x – ω · t)
where A is the amplitude, ω is the angular speed, t is the time, and x is the distance over which propagation occurred in time t, and k is called the wave number and is defined as
where λ is the wavelength, and α = ω · t is called the phase and represents an angular displacement. In SI units: x, y, and A are in meters, ω is in radians/seconds, t is in seconds, and k is in m–1 .
If a source produces waves at different times, then at the same position in space, there will be differences between the displacements because of the time delay. We call this initial phase. An example would be the case of a wave that was produced at time t before the time considered to be the origin, which would be t at 0 seconds. This means that at time 0 seconds, the wave has already reached some displacement, while the second wave starts from 0 meters displacement (see Figure 20.1).
The y (x,t) is called the harmonic wave function and is a characteristic of all periodic waves, giving the displacement and propagation for a certain moment in time (see Figure 20.2).
In the case of a three-dimensional wave, such as the water ripples in an aquarium or a lake or the waves coming from a light source, we can define a concept helpful for future developments: wavefront. Throw a stone in the water, and ripples forming from the source of perturbation will soon be visible. You can see concentric circles propagating out in space. A crosssection through water would show that the surface of the water is displaced up and down. At different positions compared to the source of perturbation, the displacement is the same: as for instance, different points are at maximum displacement (amplitude) at the same time. We call these points in phase and the circles that connect these points in phase are called wavefronts. The radius from the source to the wavefront is called a ray.
Figure 20.3 shows the wave propagation in free space, but what will happen if the space contains more than one source or obstacles? Let us answer the first part of the question. The principle of superposition says that when two or more waves are present at the same point in space and at the same time, the result of the superposition is the sum of the individual waves.
Consider two sources emitting waves and two waves meeting in a region of space. The wave diagram is shown in Figure 20.4. What is the result of the superposition of the two waves?
At each point in the figure, the two displacements are equal and opposite. Because the superposition is an algebraic sum of the two waves, the result will be zero at every point, as shown by the example on the left:
A + (–A) = 0
y1 = 20 cm · sin (1.1 · 1010 · x – 2 · t)
y2 = 23 cm · sin (0.94 · 1010 · x – 0.18 · t)
Diffraction and Huygens' Principle
If the lights are turned on in your house and you close your bedroom door, you can still tell the lights are on. How does this happen? If light only propagates in a straight line, then there should be only a small amount making its way through the keyhole or under the door. Is there something more to propagation?
The answer is diffraction, which is the bending of waves around obstacles. Bending occurs proportionally to the wavelength of the wave and is inverse proportional to the size of the obstacle. That means that for light waves (which have small wavelengths) and large openings, the ratio of the wavelength to the width (λ/w) will be very small, and the bending will not permit you to see around a corner. If we imagine the same source of waves producing some plane waves that encounter a small and a large obstacle, beyond the obstacle, diffraction will create different patterns of wave propagation, as shown in Figure 20.6. For a largesize aperture (w > > λ), no bending is produced, whereas when the two are comparable (w ~ λ), bending occurs.
The waves we see on the right of Figure 20.6 are actually composed by superimposing many waves produced by different points at the obstacle. Huygens' Principle states that every point on a wavefront produces a subset of waves that moves with the same velocity as the incident wave. At a later time, the wavefront is found to be the surface tangent to the propagating subset of waves (see Figure 20.7).
Will red light or violet light produce a broader fringe pattern on a screen? (See Figure 20.7.)
The size of the broadening pattern will be larger when more diffraction will take place. The diffraction pattern is dependent on the wavelength and the width of the opening:
Fringe pattern ~
At the same size of aperture (opening), the red light, which has a larger wavelength, will give a larger diffraction pattern, while violet (at the other end of the spectrum), with a reduced wavelength, will show a less extended fringe pattern.
Figure 20.7 not only portrays Huygens' Principle; it also shows the image formed on a screen: a strong central light followed by symmetric fringes of dark and light. Considering the earlier example with the waves that cancel each other, it will make perfect sense to say that at some points on the screen, the two or more waves present at the same time will be in opposite phase. Hence, when superposition adds the waves, the result will be zero: no light. At other positions, the resultant will be a nonzero amount and even reach a maximum when the superposition is among waves that are in phase with each other.
We call the situation when a dark fringe (no light) is obtained destructive interference, and constructive interference leads to light. The condition for destructive interference is
where m equals 1, 2, 3, and so on depending on which fringe the angle θ is considered for: the first dark fringe from the bright central spot, the second, third, and so on (see Figure 20.8).
The first dark fringe on a screen is 50 cm away from an aperture and is at a height of 5 cm above the center of the bright fringe. Find the width of the aperture (if the wavelength is 600 nm) and the angle that the first dark fringe position makes with the horizontal.
First, all units need to be converted in SI, we will then draw the diagram, and we will set our equations and solve for the unknowns.
h = 5 cm = 0.05 m
d = 50 cm = 0.50 m
λ = 600 nm = 600 · 10–9m
w = ?
θ = ?
From Figure 20.9, one can determine the angle corresponding to the first fringe:
θ = sin–l 0.125
θ = 7.2°
Using this information and the condition for the first dark fringe (m = 1), we can set a new equation:
w = 4.8 μm
A fringe pattern is not only produced when light encounters an aperture, but also when two or more parallel, thin slits are placed in front of an incident wave. A central bright fringe will form, followed by a first -order fringe on each side of the central one and so on. Both of the first -order fringes are at the same angle with respect to the central fringe, as shown in Figure 20.10.
The fringes where the result of the superposition of all the waves at a point and time is maximum are called the principal maxima (constructive interference). The fringes are less bright at other positions as well.
If the source is monochromatic (a single frequency), then the fringes are also monochromatic. If the source is a mixture of lights, then the central fringe will resemble the source, but the subsequent fringes will be split in multiple fringes, one for each of the composite colors. Therefore, a lamp producing a combination of blue and indigo as the output light after the diffraction grating will split into a higher-order, multicolor set of fringes with the blue one (λblue > λindigo) being spread farther since the corresponding angle is larger.
The condition to obtain a principal maximum is sin θ = , m = 0, 1, 2, 3,...
Diffraction gratings are characterized by the number of grooves per centimeter or per millimeter. If the number is known, then one can calculate the groove separation a.
We have studied the propagation of electromagnetic waves and should remember that they are transversal waves with an interdependent variation of electric and magnetic fields (E and B make a 90° angle and both are perpendicular at the direction of propagation). The direction of the electric field defines the direction of polarization. If a source produces waves that are all oriented such that E is directed in a single direction, we call those polarized waves (see Figure 20.11). When waves are produced such that an electric field is chaotically spread, the wave is unpolarized.
If an obstacle of a certain shape is placed in front of a source of electromagnetic waves, then the waves can be polarized. The alternation of the electrical field that sustains the magnetic field and, therefore the propagation of the wave, will not be possible in certain directions. Figure 20.12 shows a polarized wave that can propagate through the groove because the groove and the polarization are in the same plane.
And in the case where the polarization and the slit are at 90°, the light does not propagate farther than the slit (see Figure 20.13).
The light polarized can subsequently change polarization and intensity through a polarizer and an analyzer. Incoming unpolarized light will be absorbed by the polarizer in all directions other than the direction corresponding to the transmission axis of the polarizer. The average intensity of light leaving the polarizer is half the intensity of the incident light:
The light output will change further, and emerging light following the analyzer has an average intensity given by Malus' law (Etienne-Louis Malus, 1775-1812):
where is the average intensity of the outgoing light and is the average intensity of the incident polarized light, while θ is the angle between the incident and outgoing polarization of the wave.
The intensity of light is measured in watt/s2. It is proportional to the square of the electric field. Hence, an increase of E by two will increase the intensity by four (see Figure 20.14).
Consider a set of a polarizer and an analyzer, and the incident light bringing in an average intensity of 800 W /m2. What is the average intensity after the polarizer and then after the analyzer if the analyzer's transmission axis makes a 90°-angle with the axis of the polarizer?
First, we define the data and then we set the equations needed to solve the problem:
θ = 90°
Therefore, no light will be coming out of the system.
Practice problems of this concept can be found at: Diffraction and Interference Practice Questions
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