Download Wave Interference and Diffraction: A Comprehensive Guide and more Study notes Biophysics in PDF only on Docsity!
PHYSICAL OPTICS
TABLE OF CONTENTS
Interference
a. constructive and destructive
b. two source interference of light
Diffraction
a. Fresnel and Fraunhofer diffraction
b. diffraction from single slit
Interference
What is Interference?
Wave interference is the phenomenon that occurs when two waves meet while traveling along the same medium. The interference of waves causes the medium to take on a shape that results from the net effect of the two individual waves upon the particles of the medium. To begin our exploration of wave interference, consider two pulses of the same amplitude traveling in different directions along the same medium. Let's suppose that each displaced upward 1 unit at its crest and has the shape of a sine wave. As the sine pulses move towards each other, there will eventually be a moment in time when they are completely overlapped. At that moment, the resulting shape of the medium would be an upward displaced sine pulse with an amplitude of 2 units. The diagrams below depict the before and during interference snapshots of the medium for two such pulses. The individual sine pulses are drawn in red and blue and the resulting displacement of the medium is drawn in green.
Constructive Interference
Constructive interference is a type of interference that occurs at any location along the medium where the two interfering waves have a displacement in the same direction. In this case, both waves have an upward displacement; consequently, the medium has an upward displacement that is greater than the displacement of the two interfering pulses. Constructive interference is observed at any location where the two interfering waves are displaced upward. But it is also observed when both interfering waves are displaced downward. This is shown in the diagram below for two downward displaced pulses. In this case, a sine pulse with a maximum displacement of -1 unit (negative means a downward displacement) interferes with a sine pulse with a maximum displacement of -1 unit. These two pulses are drawn in red and blue. The resulting shape of the medium is a sine pulse with a maximum displacement of -2 units.
Destructive Interference
Destructive interference is a type of interference that occurs at any location along the medium where the two interfering waves have a displacement in the opposite direction. For instance, when a sine pulse with a maximum displacement of +1 unit meets a sine pulse with a maximum displacement of -1 unit, destructive interference occurs. This is depicted in the diagram below. In the diagram above, the interfering pulses have the same maximum displacement but in opposite directions. The result is that the two pulses completely destroy each other when they are completely overlapped. At the instant of complete overlap, there is no resulting displacement of the particles of the medium. This "destruction" is not a permanent condition. In fact, to say that the two waves destroy each other can be partially misleading. When it is said that the two pulses destroy each other, what is meant is that when overlapped, the effect of one of the pulses on the displacement of a given particle of the medium is destroyed or canceled by the effect of the other pulse. Waves transport energy through a medium by means of each individual particle pulling upon its nearest neighbor. When two pulses with opposite displacements (i.e., one pulse displaced up and the other down) meet at a given location, the upward pull of one pulse is balanced (canceled or destroyed) by the downward pull of the other pulse. Once the two pulses pass through each other, there is still an upward displaced pulse and a downward displaced pulse heading in the same direction that they were heading before the interference. Destructive interference leads to only a momentary condition in which the medium's displacement is less than the displacement of the largest-amplitude wave. The two interfering waves do not need to have equal amplitudes in opposite directions for destructive interference to occur. For example, a pulse with a maximum displacement of +1 unit could meet a pulse with a maximum displacement of -2 units. The resulting displacement of the medium during complete overlap is -1 unit. This is still destructive interference since the two interfering pulses have opposite displacements. In this case, the destructive nature of the interference does not lead to complete cancellation. Interestingly, the meeting of two waves along a medium does not alter the individual waves or even deviate them from their path. This only becomes an astounding behavior when it is compared to what happens when two billiard balls meet or two football players meet. Billiard balls might crash and bounce off each other and football players might crash and come to a stop. Yet two waves will meet, produce a net resulting shape of the medium, and then continue on doing what they were doing before the interference.
Two Point Source Interference
Wave interference is a phenomenon that occurs when two waves meet while traveling along the same medium. The interference of waves causes the medium to take on a shape that results from the net effect of the two individual waves upon the particles of the medium. Wave interference can be constructive or destructive in nature. Constructive interference occurs at any location along the medium where the two interfering waves have a displacement in the same direction. For example, if at a given instant in time and location along the medium, the crest of one wave meets the crest of a second wave, they will interfere in such a manner as to produce a "super-crest." Similarly, the interference of a trough and a trough interfere constructively to produce a "super-trough." Destructive interference occurs at any location along the medium where the two interfering waves have a displacement in the opposite direction. For example, the interference of a crest with a trough is an example of destructive interference. Destructive interference has the tendency to decrease the resulting amount of displacement of the medium. A defining moment in the history of the debate concerning the nature of light occurred in the early years of the nineteenth century. Thomas Young showed that an interference pattern results when light from two sources meets up while traveling through the same medium. The value of a ripple tank in the study of water wave behavior was introduced and discussed. If an object bobs up and down in the water, a series water waves in the shape of concentric circles will be produced within the water. If two objects bob up and down with the same frequency at two different points, then two sets of concentric circular waves will be produced on the surface of the water. These concentric waves will interfere with each other as they travel across the surface of the water. If you have ever simultaneously tossed two pebbles into a lake (or somehow simultaneously disturbed the lake in two locations), you undoubtedly noticed the interference of these waves. The crest of one wave will interfere constructively with the crest of the second wave to produce a large upward displacement. And the trough of one wave will interfere constructively with the trough of the second wave to produce a large downward displacement. And finally the crest of one wave will interfere destructively with the trough of the second wave to produce no displacement. In a ripple tank, this constructive and destructive interference can be easily controlled and observed. It represents a basic wave behavior that can be expected of any type of wave.
Two-Point Source Interference Patterns
The interference of two sets of periodic and concentric waves with the same frequency produces an interesting pattern in a ripple tank. The diagram at the right depicts an interference pattern produced by two periodic disturbances. The crests are denoted by the thick lines and the troughs are denoted by the thin lines. Thus, constructive interference occurs wherever a thick line meets a thick line or a thin line meets a thin line; this type of interference results in the formation of an antinode. The antinodes are denoted by a red dot. Destructive interference occurs wherever a thick line meets a thin line; this type of interference results in the formation of a node. The nodes are denoted by a blue dot. The pattern is a standing wave pattern, characterized by the presence of nodes and antinodes that are "standing still" - i.e., always located at the same position on the medium. The antinodes (points where the waves always interfere constructively) seem to be located along lines - creatively called antinodal lines. The nodes also fall along lines - called nodal lines. The two-point source interference pattern is characterized by a pattern of alternating nodal and antinodal lines. There is a central line in the pattern - the line that bisects the line segment that is drawn between the two sources is an antinodal line. This central antinodal line is a line of points where the waves from each source always reinforce each other by means of constructive interference. The nodal and antinodal lines are included on the diagram below. A two-point source interference pattern always has an alternating pattern of nodal and antinodal lines. There are however some features of the pattern that can be modified. First, a change in wavelength (or frequency) of the source will alter the number of lines in the pattern and alter the proximity or closeness of the lines. An increase in frequency will result in more lines per centimeter and a smaller distance between each consecutive line. And a decrease in frequency will result in fewer lines per centimeter and a greater distance between each consecutive line. Second, a change in the distance between the two sources will also alter the number of lines and the proximity or closeness of the lines. When the sources are moved further apart, there are more lines produced per centimeter and the lines move closer together. These two general cause-effect relationships apply to any two-point source interference pattern, whether it is due to water waves, sound waves, or any other type of wave.
Two-Point Source Light Interference Patterns
Any type of wave, whether it be a water wave or a sound wave should produce a two-point source interference pattern if the two sources periodically disturb the medium at the same frequency. Such a pattern is always characterized by a pattern of alternating nodal and antinodal lines. Before we investigate the evidence in detail, let's discuss what one might observe if light were to undergo two-point source interference. What would happen if a "crest" of one light wave interfered with a "crest" of a second light wave? And what would happen if a "trough" of one light wave interfered with a "trough" of a second light wave? And finally, what would happen if a "crest" of one light wave interfered with a "trough" of a second light wave? Whenever light constructively interferes (such as when a crest meeting a crest or a trough meeting a trough), the two waves act to reinforce one another and to produce a "super light wave." On
Young's Equation
The discussion of the interference patterns was introduced by referring to the interference of water waves in a ripple tank. All waves behave the same, whether they are water waves created by vibrating sources in a ripple tank, sound waves produced by two speakers, or light waves produced by two light sources. For water waves in a ripple tank, the resulting pattern would include locations along the water's surface where water was vibrating up and down with unusually large amplitudes (antinodes). And there would be other locations where the water was relatively undisturbed (nodes). For sound waves produced by two speakers, the interference pattern would be characterized by locations where the sound intensity was large due to constructive interference (antinodes). And there would be other locations where sound cancellation occurs and the sound intensity was relatively faint or not even heard at all (nodes). But what would be observed in a two-point source light interference pattern?
A Light Interference Pattern
As in any two-point source interference pattern, light waves from two coherent, monochromatic sources (more on coherent and monochromatic later) will interfere constructively and destructively to produce a pattern of antinodes and nodes. Light traveling through the air is typically not seen since there is nothing of substantial size in the air to reflect the light to our eyes. Thus, the pattern formed by light interference cannot be seen unless it is somehow projected onto some form of a screen or a sheet of paper. When light from the two sources is projected onto a screen, the pattern becomes quite evident. Locations where light constructively interferes corresponds to an abnormally bright spot. Locations where light destructively interferes corresponds to an abnormally dark spot. That is, the antinodes are locations where light from the two individual sources are reinforcing each other and correspond to points of brightness or maximum intensity (sometimes referred to as maxima). And the nodes are locations where light from the two individual sources are destroying each other and correspond to points of darkness or minimum intensity (sometimes referred to as minima). Red laser light passing through two narrowly spaced slits is typically used in the classroom to produce this effect. Thus, a pattern of bright red and dark fringes or bands is observed on a screen as shown in the diagram below. In the above pattern, the central bright band where light displays maximum intensity corresponds to a point on the central antinodal line. The bright bands to the right and the left of the central bright band correspond to the projection of other antinodal lines onto the screen. The dark bands correspond to the projection of the nodal lines onto the screen. Each antinodal and nodal line is assigned a number or order value (m). The red band of maximum brightness located in the center of screen (the central maximum) is assigned an order number of m =
- The other bright red bands to the left and the right of the central maximum are assigned whole number values of 1, 2, 3, ... as shown in the diagram below. The dark bands on the pattern are assigned half number values of 0.5, 1.5, 2.5, ... as shown in the diagram below.
Derivation of Young's Equation
In 1801, this experiment was performed for the first time by Thomas Young. Young expanded the mathematical model by relating the wavelength of light to observable and measurable distances. Today, an experimental setup similar to that of Thomas Young's is commonly used in a Physics classroom to repeat the experiment and to measure the wavelength of light. In the experiment as it is commonly performed today, light from a laser beam is passed through two narrowly spaced slits in a slide or sheet of paper. The light diffracts through the slits and interferes in the space beyond the slits. Thus, the slits serve as the two sources. The interference pattern is then projected onto a screen, paper or a whiteboard located several meters away. The spatial separation of nodes and antinodes on the screen is clearly seen. The most reliably measured distances in this experimental procedure are the distance from the sources to the screen, the distance between the sources, and the distance between the bright spots that appear on the screen. Thus, Thomas Young derived an equation that related the wavelength of the light to these measurable distances. The derivation, which involves relatively simple geometry, right-angle trigonometry and algebra, is repeated below.
The diagram below on the left depicts two sources labeled S1 and S2 and separated by some distance d. Point P is a point on the screen that happens to be located on some nodal or antinodal line; as such, there is an order value (m) associated with this point. Point C is the central point on the screen. The distance from point P to point C as measured perpendicular to the central antinodal line will be referred to as y. The screen is located a distance of L from the sources. In the following derivation, the wavelength of light will be related to the quantities d, m, y and L. On the diagram above, source S2 is further from point P than source S1 is. The extra distance traveled by waves from S2 can be determined if a line is drawn from S1 perpendicular to the line segment S2P. This line is drawn in the diagram on the left above; it intersects line segment S2P at point B. If point P (a bright spot on the screen) is located a great distance from the sources, then it follows that the line segment S1P is the same distance as BP. As such, the line segment S2B is simply the path difference. That is, the small distance S2B is equal to the difference in distance traveled by the two waves from their individual sources to point P on the screen. The logic is as follows: Note that step ii in the logical proof above demanded that an assumption be made: the screen must be very far away compared to the spacing between point P and the central antinodal line. That is, L >>> y. This is an assumption that underlies Young's derivation of his wavelength equation. The equation is only as valid as this assumption is true. The yellow triangle in the diagram on the left above is enlarged and redrawn in the middle of the graphic. The triangle is a right triangle with an angle theta and a hypotenuse of d. Using the sine function, it can be stated that
sine(Θ) = S 2 B / d
But since it has been previously stated that the path difference (PD) is equal to the length of the line segment S2B, the above equation can be rewritten as
sine(Θ) = PD / d
It can be further asserted that the pink triangle (Δ S1BS2) and the yellow triangle (Δ ACP) in the diagrams above are similar triangles. To prove that any two triangles are similar, one must show that they have two corresponding angles that are equal. Since the line segment PC was drawn perpendicular to the central antinodal line, it forms a 90-degree angle with the line AC. Thus, the corresponding angles S1BS2 and ACP are equal. It was shown that the path difference (PD) for any point on the pattern is equal to m • λ, where m is the order number of that point and λ is the wavelength. By substitution,
m • λ / d = y / L
As a final step in the derivation, the equation can be algebraically manipulated so that the wavelength (λ) is by itself:
λ = y • d / (m • L)
As set forth by the derivation above, the wavelength of laser light can be experimentally determined by selecting a point (referred to as point P) on a nodal and antinodal line of known order value (m) and making the following measurements:
- the distance between the slits or sources of the two light waves (d)
- the perpendicular distance from the point P to a point on the central antinodal line (y)
- the distance from point P to the sources (L)
The Importance of Coherent Light Sources
Visible light waves - those that humans can see - have an abnormally short wavelength. For instance, red light has a wavelength of about 650 nanometers. Since there are one billion nanometers in a meter, and one thousand millimeters in a meter, the wavelength of red light is less than one-thousandth of a millimeter. That's a very short wave. And being a short wave, the distance between positions of constructive interference and positions of destructive interference is very small. Thus, the effects of interference for visible light waves are difficult to observe. Complicating the task of observing the interference of visible light waves is the fact that light from the two sources must be coherent. Two light sources that maintain a constant phase difference with each other are said to be coherent light sources. Light visible to the human eye makes a complete cycle of vibration from crest to trough and back to crest in roughly 10-15 seconds. If we think of a light wave as a transverse wave pattern with crests and troughs, then a crest is typically created every 10-15 seconds. Consider two light sources producing light waves at the same frequency, but one source is creating a crest just prior to the moment in time when the other source is creating a crest. Such light sources are not at the same phase in their cycle of light production. They are said to be out of phase. Yet if they maintain the same difference in phase, they are considered coherent light sources. Even if the sources of light do not stay in step with each other, as long as the amount by which they are out of step remains the same over time, the light sources are said to be coherent.
The phenomenon of diffraction involves the spreading out of waves past openings which are on the order of the wavelength of the wave. The spreading of the waves into the area of the geometrical shadow can be modeled by considering small elements of the wavefront in the slit and treating them like point sources. If light from symmetric elements near each edge of the slit travels to the centerline of the slit, as indicated by rays 1 and 2 above, their light arrives in phase and experiences constructive interference. Light from other element pairs symmetric to the centerline also arrive in phase. Although there is a progressive change in phase as you choose element pairs closer to the centerline, this center position is nevertheless the most favorable location for constructive interference of light from the entire slit and has the highest light intensity if the Fraunhofer diffraction expression is reasonably applicable. If the conditions D >> a and D>> a2/λ are not met for this combination of slit width and screen distance, the Fresnel diffraction result may not have maximum intensity on the centerline. The first minimum in intensity for the light through a single slit can be visualized in terms of rays 3 and 4. An element at one edge of the slit and one just past the centerline are chosen, and the condition for minimum light intensity is that light from these two elements arrive 180° out of phase, or a half wavelength different in pathlength. If those two elements suffer destructive interference, then choosing additional pairs of identical spacing which progress downward across the slit will give destructive interference for all those pairs and therefore an overall minimum in light intensity. One of the characteristics of single slit diffraction is that a narrower slit will give a wider diffraction pattern as illustrated below, which seems somewhat counter-intuitive. One way to visualize it is to consider that rays 3 and 4 must reach one half wavelength difference in light pathlength, and if the slit is narrower, it will take a greater angle of the rays to achieve that difference. The diffraction patterns were taken with a helium-neon laser and a narrow single slit. The slit widths used were on the order of 100 micrometers, so their widths were 100 times the laser wavelength or more. A slit width equal to the wavelength of the laser light would spread the first minimum out to 90° so that no minima would be observed. The relationships between slit width and the minima and maxima of diffraction can be explored in the single slit calculation. These Fraunhofer single slit diffraction images were taken with laser sources, which easily give you the conditions under which Fraunhofer diffraction applies, i.e., the laser beam produces an essentially infinite effective object distance. The Fraunhofer diffraction conditions can also be produced by using two lenses, with a source slit placed at the focal length of the first lens, and the viewing screen placed past the second lens at its focal length. This arrangement is shown below. The lens arrangement forces the beam to be parallel to the axis at the position of the diffracting slit.
Double Slit
The pattern formed by the interference and diffraction of coherent light is distinctly different for a single and double slit. The single slit intensity envelope is shown by the dashed line and that of the double slit for a particular wavelength and slit width is shown by the solid line. The photographs of the single and double slit patterns produced by a helium-neon laser show the qualitative differences between the patterns produced. You can see that the drawing is not to the same scale as the photographs, but the breaking up of the broad maxima of the single slit pattern into more closely spaced maxima is evident. The number of bright maxima within the central maximum of the single-slit pattern is influenced by the width of the slit and the separation of the double slits.
Multiple slits
Grating Intensity Comparison
The grating intensity expression gives a peak intensity which is proportional to the square of the number of slits illuminated. Increasing the number of slits not only makes the diffraction maximum sharper, but also much more intense. If a 1 mm diameter laser beam strikes a 600 line/mm grating, then it covers 600 slits and the resulting line intensity is 90,000 x that of a double slit. As the intensity increases, the diffraction maximum becomes narrower as well as more intense. When you have 600 slits, the maxima are very sharp and bright and permit high-resolution separation of the maxima for different wavelengths. Such a multiple-slit is called a diffraction grating.
Fresnel Diffraction: Single Slit
The more accurate Fresnel treatment of the single slit may give a pattern which is similar in appearance to that of the Fraunhofer single slit except that the minima are not exactly zero. In these circumstances the Fraunhofer diffraction from the slit is a satisfactory approximation of the optical phenomenon. But for circumstances where the distance to the screen is not sufficiently large compared to the width of the diffracting slit, the Fresnel diffraction pattern may be dramatically different.
Comparison: Fraunhofer & Fresnel Slit
The more accurate Fresnel treatment of the single slit may give a pattern which is similar in appearance to that of the Fraunhofer single slit except that the minima are not exactly zero. In these circumstances the Fraunhofer diffraction from the slit is a satisfactory approximation of the optical phenomenon. But for circumstances where the distance to the screen is not sufficiently large compared to the width of the diffracting slit, the Fresnel diffraction pattern may be dramatically different.
When a wave hits an obstacle it does not simply go straight past, it bends round the obstacle. The same type of effect occurs at a hole - the waves spread out the other side of the hole. This phenomenon is known as diffraction and examples of the diffraction of plane waves are shown in Figure 1. The effects of diffraction are much more noticeable if the size of the obstacle is small (a few wavelengths across), while a given size of obstacle will diffract a wave of long wavelength more than a shorter one. Diffraction can be easily demonstrated with sound waves or microwaves. It is quite easy to hear a sound even if there is an obstacle in the direct line between the source and your ears. By using the 2.8 cm microwave apparatus owned by many schools very good diffraction effects may be observed with obstacles a few centimetres across. One of the most powerful pieces of evidence for light being some form of wave motion is that it also shows diffraction. The problem with light, and that which led Newton to reject the wave theory is that the wavelength is very small and therefore diffraction effects are hard to observe. You can observe the diffraction of light, however, if you know just where to look. The coloured rings round a street light in frosty weather, the coloured bands viewed by reflection from a record and the spreading of light round your eyelashes are all diffraction effects. Looking through the material of a stretched pair of tights at a small torch bulb will also show very good diffraction. A laser will also show good diffraction effects over large distances because of the coherence of laser light. Diffraction is essentially the effect of removing some of the information from a wave front; the new wave front will be altered by the obstacle or aperture. Huygens' theory explained this satisfactorily. Grimaldi first recorded the diffraction of light in 1665 but the real credit for its scientific study must go to Fresnel, Poisson and Arago, working in the late eighteenth and early nineteenth centuries. Fresnel and Fraunhofer diffraction We can define two distinct types of diffraction: (a) Fresnel diffraction is produced when light from a point source meets an obstacle, the waves are spherical and the pattern observed is a fringed image of the object. (b) Fraunhofer diffraction occurs with plane wave-fronts with the object effectively at infinity. The pattern is in a particular direction and is a fringed image of the source. Fresnel diffraction Fresnel diffraction can be observed with the minimum of apparatus but the mathematics are complex. We will therefore only treat it experimentally here. If a razor blade is placed between the observer and a point source of monochromatic light, dark and bright diffraction fringes can be seen in the edges of the shadow. The same effects can be produced with a pinhead, when a spot of light will be seen in the centre of the shadow. Fresnel was unhappy about Newton's explanation of diffraction in terms of the attraction of the light particles by the particles of the solid, because diffraction was found to be independent of the density of the obstacle: a spider's web, for example, gave the same diffraction pattern as a platinum wire of the same thickness. The prediction and subsequent discovery of a bright spot within the centre of the shadow of a small steel ball was final proof that light was indeed a wave motion. If the intensity of light is plotted against distance for points close to the shadow edge results like those shown in Figure 2 will be obtained. Fresnel diffraction with a double slit will produce two single slit patterns superimposed on one another. This is exactly what happens in the Young's slit experiment: the diffraction effects are observed as well as those due to the interference of the two sets of waves.
