Diffraction and Interference Study Guide (page 2)
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).
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