Course: Physics 1. Module 3: Optics and Wave Phenomena

Transcription

1 MINISTRY OF EDUCATION AND TRAINING NONG LAM UNIVERSITY FACULTY OF FOOD SCIENCE AND TECHNOLOGY Course: Physics 1 Module 3: Optics and Wave Phenomena Instructor: Dr. Son Thanh Nguyen Academic year:

2 Contents Module 3: Optics and wave phenomena 3.1. Wave review 1) Description of a wave 2) Transerve waves and longitudinal waves 3) Mathematical description of a traveling (propagating) wave with constant amplitude 4) Electromagnetic waves 5) Spherical and plane waves 3.2. Interference of sound waves and light waves 1) Interference of sinusoidal waves Coherent sources 2) Interference of sound waves 3) Interference of light waves 3.3. Diffraction and spectroscopy 1) Introduction to diffraction 2) Diffraction by a single narrow slit - Diffraction gratings 3) Spectroscopy: Dispersion Spectroscope Spectra 3.4. Applications of interference and diffraction 1) Applications of interference 2) Applications of diffraction 3.5. Wave-particle duality of light and particles 1) Photoelectric effect Einstein s photon concept 2) Electromagnetic waves and photons 3) Wave-particle duality De Broglie s postulate Physic 1 Module 3: Optics 2

3 3.1. Wave review 1) Description of a propagating wave Figure 23: Representation of a typical wave, showing its direction of motion, wavelength, and amplitude. Simply stated, a wave is a way in which energy is transferred from place to place without physical movement of material from one location to another. In wave motion, the energy is carried by a disturbance of some sort. This disturbance, whatever its nature, occurs in a distinctive repeating pattern. Ripples on the surface of a pond, sound waves in air, and electromagnetic waves in space, despite their many obvious differences, all share this basic defining property. In other words, wave is a periodic disturbance that travels from one place to another without actually transporting any matter. The source of all waves is something that is vibrating, moving back and forth at a regular, and usually fast rate. We must distinguish between the motion of particles of the medium through which the wave is propagating and the motion of the wave pattern through the medium, or wave motion. The particles of the medium vibrate at fixed positions; the wave progresses through the medium. Familiar examples of waves are waves on a surface of water; waves on a stretched string; sound waves; light and other forms of electromagnetic radiation. While a mechanical wave such as a sound wave exists in a medium, waves of electromagnetic radiation including light can travel through vacuum, that is, without any medium. Periodic waves are characterized by crests (highs) and troughs (lows), as shown in Figure 23. Within a wave, the phase of a vibration of the medium s particle (that is, its position within the vibration cycle) is different for adjacent points in space because the wave reaches these points at different times. Waves travel and transfer energy from one point to another, often with little or no permanent displacement of the particles of the medium (that is, with little or no associated mass transport); instead there are oscillations (vibrations) around almost fixed locations. Physic 1 Module 3: Optics 3

4 2) Transverse and longitudinal waves In terms of the direction of particles s vibrations and that of the wave propagation, there are two major kinds of waves: transverse waves and longitudinal waves. Transverse waves are those with particles s vibrations perpendicular to the wave's direction of travel; examples include waves on a stretched string and electromagnetic waves. Longitudinal waves are those with particles s vibrations along the wave's direction of travel; examples include sound waves. Figure 24: When an object bobs up and down on a ripple in a pond, it experiences an elliptical trajectory because ripples are not simple transverse sinusoidal waves. Apart from transverse waves and longitudinal waves, ripples on the surface of a pond are actually a combination of transverse and longitudinal waves; therefore, the points on the water surface follow elliptical paths, as shown in Figure 24. 3) Mathematical description of a traveling (propagating) wave with constant amplitude Transverse waves are probably the most important waves to understand in this module; light is also a transverse wave. We will therefore start by studying transverse waves in a simple context: waves on a stretched string. As mentioned earlier, a transverse, propagating wave is a wave that consists of oscillations of the medium s particles perpendicular to the direction of wave propagation or energy transfer. If a transverse wave is propagating in the positive x-direction, the oscillations are in up and down directions that lie in the yz-plane. From a mathematical point of view, the most primitive or fundamental wave is harmonic (sinusoidal) wave which is described by the wave function u(x, t) = Asin(kx ωt) (47) where u is the displacement of a particular particle of the medium from its midpoint, A the amplitude of the wave, k the wave number, ω the angular frequency, and t the time. In the illustration given by Figure 23, the amplitude is the maximum vertical distance between the baseline and the wave or the maximum departure of the wave from the undisturbed state. The units of the amplitude depend on the type of wave - waves on a string have an amplitude expressed as a distance (meters), sound waves as pressure (pascals), and electromagnetic waves as an magnitude of the electric field (volts/meter). The amplitude may be constant or may vary with time and/or position. The form of the variation of amplitude is called the envelope of the wave. Physic 1 Module 3: Optics 4

5 The period T is the time for one complete cycle for an oscillation. The frequency f (also frequently denoted as ν) is the number of periods per unit time (one second) and is measured in hertz. T and f are related by f = 1 T (48) In other words, the frequency and period of a wave are reciprocals of each other. The frequency is equal to the number of crests or cycles passing any given point per unit time (a second). The angular frequency ω represents the frequency in terms of radians per second. It is related to the frequency f by ω = 2πf (49) There are two velocities that are associated with waves. The first is the phase velocity, v p or v, which gives the rate at which the wave propagates, is given by v = k ω (50) The second is the group velocity, v g, which gives the velocity at which variations in the shape of the wave's amplitude propagate through space. This is the rate at which information can be transmitted by the wave. It is given by v g = ω k (51) The wavelength (denoted as λ) is the distance between two successive crests (or troughs) of a wave, as shown in Figure 22. This is generally measured in meters; it is also commonly measured in nanometers for the optical part of the electromagnetic spectrum. The wavelength is related to the period (or frequency) and speed of a wave (phase velocity) by the equation λ = vt = v/f (52) For example, a radio wave of wavelength 300 m traveling at 300 million m/s (the speed of light) has a frequency of 1 MHz. The wavenumber k is associated with the wavelength by the relation k = 2π λ (53) Example: Thomas attaches a stretched string to a mass that oscillates up and down once every half second, sending waves out across the string. He notices that each time the mass reaches the maximum positive displacement of its oscillation, the last wave crest has just reached a bead attached to the string 1.25 m away. What are the frequency, wavelength, and speed of the waves? (Ans. f = 2 Hz, λ = 1.25 m, v = 2.5 m/s) Physic 1 Module 3: Optics 5

6 4) Electromagnetic waves As described earlier, a transverse, moving wave is a wave that consists of oscillations perpendicular to the direction of energy transfer. If a transverse wave is moving in the positive x-direction, the oscillations are in up and down directions that lie in the yz-plane. Figure 25: Electric and magnetic fields vibrate perpendicular to each other. Together they form an electromagnetic wave that moves through space at the speed of light c. Electromagnetic (EM) waves including light behave in the same way as other waves, although it is harder to see. Electromagnetic waves are also twodimensional transverse waves. This twodimensional nature should not be confused with the two components of an electromagnetic wave, the electric and magnetic field components, which are shown in shown in Figure 25. Each of these fields, the electric and the magnetic, exhibits two-dimensional transverse wave behavior, just like the waves on a string, as shown in Figure 25. Figure 26: Spherical waves emitted by a point source. The circular arcs represent the spherical wave fronts that are concentric with the source. The rays are radial lines pointing outward from the source, perpendicular to the wave fronts. A light wave is an example of an electromagnetic wave which is shown in Figure 25. In vacuum light propagate with phase speed: v = c = 3 x 10 8 m/s. The term electromagnetic just means that the energy is carried in the form of rapidly fluctuating electric and magnetic fields. Visible light is the particular type of electromagnetic wave (radiation) to which our human eyes happen to be sensitive. But there is also invisible electromagnetic radiation, which goes completely undetected by our eyes. Radio, infrared, and ultraviolet waves, as well as x rays and gamma rays, all fall into this category. Physic 1 Module 3: Optics 6

7 5) Spherical and plane waves If a small spherical body, considered as a point, oscillates so that its radius varies sinusoidally with time, a spherical wave is produced, as shown in Figure 26. The wave moves outward from the source in all directions, at a constant speed if the medium is uniform. Due to the medium s uniformity, the energy in a spherical wave propagates equally in all directions. That is, no one direction is preferred over any other. It is useful to represent spherical waves with a series of circular arcs concentric with the source, as shown in Figure 26. Each arc represents a surface over which the phase of the wave is constant. We call such a surface of constant phase a wave front. The distance between adjacent wave fronts equals the wavelength λ. The radial lines pointing outward from the source and perpendicular to the wave fronts are called rays. Now consider a small portion of a wave front far from the source, as shown in Figure 27. In this case, the rays passing through the wave front are nearly parallel to one another, and the wave front is very close to being planar. Therefore, at distances from the source that are great compared with the wavelength, we can approximate a wave front with a plane. Any small portion of a spherical wave front far from its source can be considered a plane wave front. Figure 28 illustrates a plane wave propagating along the x axis, which means that the wave Figure 27: Far away from a point source, the wave fronts are nearly parallel planes, and the rays are nearly parallel lines perpendicular to these planes. Hence, a small segment of a spherical wave is approximately a plane wave. fronts are parallel to the yz plane. In this case, the wave function depends only on x and t and has the form u(x, t) = Asin(kx - ωt) (54) That is, the wave function for a plane wave is identical in form to that for a one-dimensional traveling wave (equation 47). The intensity is the same at all points on a given wave front of a plane wave. In other words, a plane wave have wave fronts that are planes parallel to each other, rather than Figure 28: A representation of a plane wave moving in the positive x direction with a speed v. The wave fronts are planes parallel to the yz plane. Physic 1 Module 3: Optics 7

8 spheres of increasing radius Interference of sound waves and light waves Interference of waves What happens when two waves meet while they travel through the same medium? What affect will the meeting of the waves have upon the appearance of the medium? These questions involving the meeting of two or more waves in the same medium pertain to the topic of wave interference. Wave interference is a phenomenon which occurs when two waves of the same frequency and of the same type (both are transverse or longitudinal) meet while traveling along the same medium. The interference of waves causes the medium to take on a shape which results from the net effect of the two individual waves upon the particles of the medium. In other words, interference is the ability of two or more waves to reinforce or partially cancel each other. Physic 1 Module 3: Optics 8 Figure 29: Depicting the snapshots of the medium for two pulses of the same amplitude (both upward) before and during interference; the interference is constructive. To begin our exploration of wave interference, consider two sine pulses of the same amplitude traveling in different directions in the same medium. Suppose that each is displaced upward 1 unit at its crest and has the shape of a sine wave. As the sine pulses move toward 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 shown in Figure 29 depict the snapshots of the medium for two such pulses before and during interference. The individual sine pulses are drawn in red and blue, and the resulting displacement of the medium is drawn in green. This type of interference is called constructive interference. Constructive interference is a type of interference which occurs at any location in the medium where the two interfering waves have a displacement in the same direction and their crests or troughs exactly coincide. The net effect is that the two wave motions reinforce each other, resulting in a wave of greater amplitude. In the case mentioned above, both waves have an upward displacement; consequently, the medium has an upward displacement which is greater than the displacement of either interfering pulse. Constructive interference is observed at any location where the two interfering waves are displaced upward. But it is also observed when both interfering Figure 30: Depicting the snapshots of the medium for two pulses of the same amplitude (both downward) before and during interference; the interference is constructive.

9 waves are displaced downward. This is shown in Figure 30 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 again drawn in red and blue. The resulting shape of the medium is a sine pulse with a maximum displacement of -2 units. Destructive interference is a type of interference which occurs at any location in the medium where the two interfering waves have displacements 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 shown in Figure 31. In Figure 31, 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. When two pulses with opposite displacements (i.e., one pulse displaced up and the Figure 31: Depicting the snapshots of the medium for two pulses of the same amplitude (one upward and one downward) before and during interference; the interference is destructive. 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. 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, as shown in Figure 32. Figure 32: Depicting the before and during interference snapshots of the medium for two pulses of different amplitudes (one upward, +1 unit and one downward, -2 unit); the interference is destructive. The task of determining the shape of the resultant wave demands that the principle of superposition is applied. The principle of superposition is stated as follows: When two waves interfere, the resulting displacement of the medium at any location is the algebraic sum of the displacements of the individual waves at that same location. In the cases mentioned above, the summing of the individual displacements for locations of complete overlap was easy and given in the below table. Physic 1 Module 3: Optics 9

10 Maximum displacement of Pulse 1 Maximum displacement of Pulse ) Interference of sinusoidal waves Coherent sources Mathematics of two-point source interference Physic 1 Module 3: Optics 10 Maximum resulting displacement We already found that the adding together of two mechanical waves can be constructive or destructive. In constructive interference, the amplitude of the resultant wave is greater than that of either individual wave, whereas in destructive interference, the resultant amplitude is less than that of either individual wave. Light waves also interfere with each other. Fundamentally, all interference associated with light waves arises when the electromagnetic fields that constitute the individual waves combine. Conditions for interference For sustained interference in waves to be observed, the following conditions must be met: The sources must maintain a constant phase with respect to each other. The sources should of a single wavelength (or frequency). Such wave sources are termed coherent sources. We now describe the characteristics of coherent sources. As we saw when we studied mechanical waves, two sources (producing two traveling waves) are needed to create interference. In order to produce a stable interference pattern, the individual waves must maintain a constant phase relationship with one another. As an example, the sound waves emitted by two side-by-side loudspeakers driven by a single amplifier can interfere with each other because the two speakers are coherent - that is, they respond to the amplifier in the same way at the same time. A common method for producing two coherent sources is to use one monochromatic source to generate two secondary sources. For example, a popular method for producing two coherent light sources is to use one monochromatic source to illuminate a barrier containing two small openings (usually in the shape of slits). The light emerging from the two slits is coherent because a single source produces the original light beam and the two slits serve only to separate the original beam into two parts (which, after all, is what was done to the sound signal from the side-by-side loudspeakers). Consider two separate waves propagating from two coherent sources located at O 1 and O 2. The waves meet at P, and according to the principle of superposition, the resultant vibration at P is given by u P = u 1 + u 2 = Asin(kx 1 ωt) + Asin(kx 2 ωt) (55) where x 1 = O 1 P and x 2 = O 2 P are the wave paths (distances traveled) from O 1 and O 2 to P, respectively.

11 For the sake of simplicity, we have assumed A 1 = A 2 = A. Using the trigonometric identity: sinα + sinβ = 2sin{(α+β)/2}cos{(α-β)/2} (56), from equation (55) we have u P = 2Acos{k(x 2 x 1 )/2}sin{k(x 1 + x 2 )/2 - ωt} (57) From equation (57), we see that the amplitude A M of the resultant vibration (resultant amplitude) at the point P is given by A P = 2Acos {k(x 2 x 1 )/2} (58) According to equation (58), A P is time independent and depends only on the path difference, Δx, of the two wave components Δx = x 2 x 1 (59) From equations (53), (58) and (59), we can easily see the following cases: Case 1: Δx = x 2 x 1 = n2π/k = nλ where n = 0, ±1, ±2, or the path difference is zero or some integer multiple of wavelength. We have A P = 2A. The amplitude of the resultant wave is 2A - twice the amplitude of either individual wave. In this case, the interfereing waves are said to be everywhere in phase and thus interfere constructively. There is a constructive interference at P. Case 2: Δx = x 2 x 1 = (n + 0,5)π/k = (2n + 1)λ/2 where n = 0, ±1, ±2, or the path difference is odd multiple of half wavelength. We have A P = 0. The resultant wave has zero amplitude. In this case, the interfereing waves are exactly 180 o out of phase and thus interfere destructively. There is a destructive interference at P. 2) Interference of sound waves One simple device for demonstrating interference of sound waves is illustrated in Figure 33. Sound from a loudspeaker S is sent into a tube at point P, where there is a T-shaped junction. Figure 33: An acoustical system for demonstrating interference of sound waves. A sound wave from the speaker (S) propagates in the tube and splits into two parts at point P. The two waves, which superimpose at the opposite side, are detected at the receiver (R). The upper path length r 2 can be varied by sliding the upper section. Physic 1 Module 3: Optics 11

12 Half of the sound power travels in one direction, and half travels in the opposite direction. Thus, the sound waves that reach the receiver R can travel along either of the two paths. The distance along any path from speaker to receiver is called the path length r. The lower path length r 1 is fixed, but the upper path length r 2 can be varied by sliding a U-shaped tube, which is similar to that on a slide trombone. When the path difference is either zero or some integer multiple of the wavelength λ (that is r 2 r 1 = nλ, where n = 0, ±1, ±2,...), the two waves reaching the receiver at any instant are in phase and interfere constructively. For this case, a maximum in the sound intensity is detected at the receiver. If the path length r 2 is adjusted such r 2 r 1 = (n + 1/2)λ, where n = 0, ±1, ±2,..., the two waves are exactly π rad, or 180, out of phase at the receiver and hence cancel each other. In this case of destructive interference, no sound is detected at the receiver. 3) Interference of light waves Two-point source light interference patterns Any type of wave, whether it is 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. Let's discuss what one might observe if light were to undergo two-point source interference. What will happen if a "crest" of one light wave interferes with a "crest" of a second light wave? And what will happen if a "trough" of one light wave interferes with a "trough" of a second light wave? And finally, what will happen if a "crest" of one light wave interfered with a "trough" of a second light wave? Whenever light waves constructively interfere (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 an enhanced light wave. On the other hand, whenever light waves destructively interfere (such as when a crest meets a trough), the two waves act to destroy each other and produce no light wave. Thus, the two-point source interference pattern would still consist of an alternating pattern of antinodal lines and nodal lines. For light waves, the antinodal Figure 34: Schematic diagram of Young s double-slit experiment. Two slits behave as coherent sources of light waves that produce an interference pattern on the viewing screen (drawing not to scale). Physic 1 Module 3: Optics 12

13 lines are equivalent to bright lines, and the nodal lines are equivalent to dark lines. If such an interference pattern could be created by two light sources and projected onto a screen, then there ought to be an alternating pattern of dark and bright bands on the screen. And since the central line in such a pattern is an antinodal line, the central band on the screen ought to be a bright band. YOUNG S DOUBLE-SLIT EXPERIMENT In 1801, Thomas Young successfully showed that light does produce a two-point source interference pattern. In order to produce such a pattern, monochromatic light must be used. Monochromatic light is light of a single color; by use of such light, the two sources will vibrate with the same frequency. It is also important that the two light waves be vibrating in phase with each other; that is, the crest of one wave must be produced at the same precise time as the crest of the second wave. (These waves are often referred to as coherent light waves.) As expected, the use of a monochromatic light source and pinholes to generate in-phase light waves resulted in a pattern of alternating bright and dark bands on the screen. A typical appearance of the pattern is shown in Figure 35. To accomplish this, Young used a single light source (primary source) and projected the light onto two very narrow slits, as shown in Figure 34. The light from the source will then diffract through the slits, and the pattern can be projected onto a screen. Since there is only one source of light, the set of two waves which emanate from the slits will be in phase with each other. As a result, these two slits, denoted as S 1 and S 2, serve as a pair of coherent light sources. The light waves from S 1 and S 2 produce on a viewing screen a visible Figure 35: A typical pattern from a two-slit experiment of interference. pattern of bright and dark parallel bands called fringes, as shown in Figure 35. When the light from S 1 and that from S 2 both arrive at a point on the screen such that constructive interference occurs at that location, a bright fringe appears. When the light from the two slits combines destructively at any location on the screen, a dark fringe results. Physic 1 Module 3: Optics 13

14 We can describe Young s experiment quantitatively with the help of Figure 36. The viewing screen is located a perpendicular distance L from the doubleslitted barrier. S 1 and S 2 are separated by a distance d, and the source is monochromatic. To reach any arbitrary point P, a wave from the lower slit travels farther than a wave from the upper slit by a distance d sin θ. This distance is called the path difference δ (lowercase Greek delta). If we assume that two rays, S 1 P and S 2 P, are parallel, which is approximately true because L is much greater than d, then δ is given by Figure 36: Geometric construction for describing Young s doubleslit experiment (not to scale). δ = S 2 P S 1 P = r 2 r 1 = d sin θ (60) where d = S 1 S 2 is the distances between the two coherent light sources (i.e., the two slits). If δ is either zero or some integer multiple of the wavelength, then the two waves are in phase at point P and constructive interference results. Therefore, the condition for bright fringes, or constructive interference, at point P is where n = 0, ±1, ±2,. δ = r 2 r 1 = nλ (61) The number n in equation (61) is called the order number. The central bright fringe at θ = 0 (n = 0) is called the zeroth-order maximum. The first maximum on either side, where n = ±1, is called the first-order maximum, and so forth. When δ is an odd multiple of λ/2, the two waves arriving at point P are 180 out of phase and give rise to destructive interference. Therefore, the condition for dark fringes, or destructive interference, at point P is where n = 0, ±1, ±2,... δ = r 2 r 1 = (n + 1/2)λ (62) Physic 1 Module 3: Optics 14

15 It is useful to obtain expressions for the positions of the bright and dark fringes measured vertically from O to P. In addition to our assumption that L >> d, we assume that d >> λ. These can be valid assumptions because in practice L is often of the order of 1 m, d a fraction of a millimeter, and λ a fraction of a micrometer for visible light. Under these conditions, θ is small; thus, we can use the approximation sin θ tan θ. Then, from triangle OPQ in Figure 36, we see that y = OP = L tan θ L sin θ (63) From equations (60), (61) and (63), we can prove that the positions of the bright fringes measured from O are given by the expression y bright = n λl d (64) Similarly, using equations (60), (62) and (63), we find that the dark fringes are located at λl y dark = (n + 1/2) d (65) As we demonstrate in the following example, Young s double-slit experiment provides a method for measuring the wavelength of light. In fact, Young used this technique to do just that. Additionally, the experiment gave the wave model of light a great deal of credibility. It was inconceivable that particles of light coming through the slits could cancel each other in a way that would explain the dark fringes. As a result, the light interference show that light is of wave nature. Example: A viewing screen is separated from a double-slit source by 1.2 m. The distance between the two slits is mm. The second-order bright fringe is 4.5 cm from the center line. (a) Determine the wavelength of the light. (Ans. λ = 560 nm) (b) Calculate the distance between two successive bright fringes. (Ans cm) Physic 1 Module 3: Optics 15

16 Intensity distribution of the double-slit interference pattern So far we have discussed the locations of only the centers of the bright and dark fringes on a distant screen. We now direct our attention to the intensity of the light at other points between the positions of constructive and destructive interference. In other words, we now calculate the distribution of light intensity associated with the double-slit interference pattern. Figure 37: Light intensity versus δ = d sin θ for a doubleslit interference pattern when the viewing screen is far from the slits (L >> d). Again, suppose that the two slits represent coherent sources of sinusoidal waves such that the two waves from the slits have the same frequency f and a constant phase difference. Recall that the intensity of a light wave, I, is proportional to the square of the resultant electric field magnitude at the point of interest, we can show that (see pages 1191 and 1192, Halliday s book). πd I = I max cos ( y) λl 2 (66) where I max is the maximum intensity on the screen, and the expression represents the time average. Constructive interference, which produces light intensity maxima, occurs when the quantity πy/λl is an integral multiple of π, corresponding to y = (λl/d)n. This is consistent with equation (64). Physic 1 Module 3: Optics 16

17 A plot of light intensity versus δ = d sinθ is given in Figure 37. Note that the interference pattern consists of equally spaced fringes of equal intensity. Remember, however, that this result is valid only if the slit-toscreen distance is much greater than the slit separation (L >> d), and only for small values of θ Diffraction and spectroscopy 1) Introduction to diffraction Diffraction is the deflection, or "bending," of a wave as it passes a corner or moves through a narrow gap. For any wave, the amount of diffraction is proportional to the ratio of the wavelength to the width of the gap. The longer the wavelength and/or the smaller the gap, the greater the angle through which the wave is diffracted. Thus, visible light, with its extremely short wavelengths, shows perceptible diffraction only when passing through very narrow openings. (The effect is much more noticeable for sound Figure 38: Diffraction of a light wave: (a) If radiation were composed of rays or particles moving in perfectly straight lines, no bending would occur as a beam of light passed through a circular hole in a barrier, and the outline of the hole, projected onto a screen, would have perfectly sharp edges. (b) In fact, light is diffracted through an angle that depends on the ratio of the wavelength of the wave to the size of the gap. The result is that the outline of the hole becomes "fuzzy," as shown in this actual photograph of the diffraction pattern. waves, however - no one thinks twice about our ability to hear people even when they are around a corner and out of our line of sight.) Diffraction is normally taken to refer to various phenomena which occur when a wave encounters an obstacle whose size is comparable to the wavelength. It is described as the apparent bending of waves around small obstacles and the spreading out of waves past small openings. Diffraction occurs with all waves, including sound waves, water waves, and electromagnetic waves such as visible light, x-rays, and radio waves. Diffraction is a property that distinguishes between wave-like and particle-like behaviors. A slit of infinitesimal width which is illuminated by light diffracts the light into a series of circular waves and the wavefront which emerges from the slit is a cylindrical wave of uniform intensity. The light at a given angle is made up of contributions from each of these point sources, and if the relative phases of these contributions vary by more than 2π, we expect to find minima and maxima in the diffracted light. Physic 1 Module 3: Optics 17

18 The effects of diffraction can be readily seen in everyday life. The most colorful examples of diffraction are those involving light; for example, the closely spaced tracks on a CD or DVD act as a diffraction grating to form the familiar rainbow pattern we see when looking at a disk. All these effects are a consequence of the fact that light is a wave. Diffraction arises because of the way in which waves propagate; this is described by the Huygens Fresnel principle. This principle states that Eeach point of an advancing wave front is in fact the center of a fresh disturbance and the source of a new train of waves; and that the advancing wave as a whole may be regarded as the sum of all the secondary waves arising from points in the medium already traversed. The propagation of a wave can be visualized by considering every point on a wavefront as a point source for a secondary radial wave. The subsequent propagation and addition of all these radial waves form the new wavefront, as shown in Figure 38. When waves are added together, their sum is determined by the relative phases as well as the amplitudes of the individual waves, an effect which is often known as wave interference. The resultant amplitude of the waves can have any value between zero and the sum of the individual amplitudes. Hence, diffraction patterns usually have a series of maxima and minima (see Figure 38b). To determine the form of a diffraction pattern, we must determine the phase and amplitude of each of the Huygens wavelets at each point in space and then find the sum of these waves. There are various analytical models which can be used to do this including the Fraunhoffer diffraction equation for the far field and the Fresnel diffraction equation for the near field. As a result, diffraction effects are classified into either Fresnel or Fraunhofer types. Fresnel diffraction is concerned mainly with what happens to light in the immediate neighborhood of a diffracting object or aperture, so is only of concern when the illumination source is close by. Fraunhofer diffraction is the light-spreading effect of an aperture when the aperture (or object) is lit by plane waves, i.e., waves that effectively come from a source that is infinitely far away. Because of Fraunhofer diffraction, a telescope can never form a perfect image. A point-like source, for example, will be seen as a small disk surrounded by a series of rings; a thin line on a planet will become widened into a band, which decreases in intensity on both sides. The only way to overcome the limitations of diffraction is to use a telescope of larger aperture. Diffraction is set to work in diffraction gratings. Here, light passed through a series of very accurately ruled slits. Gratings are ruled from 70 lines/mm (for infrared work) to 1800 lines/mm (for ultraviolet work). 2) Diffraction by a single narrow slit Single-slit diffraction This is an attempt to more clearly visualize the nature of single-slit diffraction. 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. Physic 1 Module 3: Optics 18

19 Figure 39: (a) Fraunhofer diffraction pattern of a single slit. The pattern consists of a central bright fringe flanked by much weaker maxima alternating with dark fringes (drawing not to scale). (b) Photograph of a single-slit Fraunhofer diffraction pattern. In general, diffraction occurs when waves pass through small openings, around obstacles, or past sharp edges, as shown in Figure 39. When an opaque object is placed between a point source of light and a screen, no sharp boundary exists on the screen between a shadowed region and an illuminated region. The illuminated region above the shadow of the object contains alternating light and dark fringes. Such a display is called a diffraction pattern (see Figure 38.3, Halliday s book, page 1213). Figure 38.3 shows a diffraction pattern associated with the shadow of a penny. In this module we restrict our attention to Fraunhofer diffraction, which occurs, for example, when all the rays passing through a narrow slit are approximately parallel to one another (a plane wave). This can be achieved experimentally either by placing the screen far from the opening used to create the diffraction or by using a converging lens to focus the rays once they pass through the opening, as shown in Figure 39a. A bright fringe is observed along the axis at θ = 0, with alternating dark and bright fringes occurring on either side of the central bright one. Figure 39b is a photograph of a single-slit Fraunhofer diffraction pattern. We can find the angle at which a first minimum is obtained in the diffracted light by the following reasoning. The light from a source located at the top edge of the slit interferes destructively with a source located at the middle of the slit, when the path difference between them is equal to λ/2. Similarly, the source just below the top of the slit will interfere destructively with the source located just below the middle of the slit at the same angle. We can continue this reasoning along the entire height of the slit to conclude that the condition for destructive interference for the entire slit is the same as the condition for destructive interference Physic 1 Module 3: Optics 19

20 between two narrow slits a distance apart that is half the width of the slit (see section 3.2.3). The path difference is given by (a sinθ)/2 so that the minimum intensity occurs at an angle θ min given by a sin θ min = λ (67) where a is the width of the slit. A similar argument can be used to show that if we imagine the slit to be divided into four, six eight parts, etc, minima are obtained at angles θ n given by a sin θ n = nλ (68) where n is an integer greater than zero. The intensity distribution for a Fraunhofer diffraction pattern from a single slit of width a shown in Figure 40. It should be noted that this analysis applies only to the far field, that is at a distance much larger than the width of the slit. Diffraction gratings Diffraction grating is an optical device used to disperse light into a spectrum. It is ruled with closely-spaced, fine, parallel grooves, typically several thousand per cm, that produce interference patterns in a way that separates all the components of the incoming light, as shown in Figure 41. A diffraction grating can be used as the main dispersing element in a spectroscope (see the next section). Figure 40: Intensity distribution for a Fraunhofer diffraction pattern from a single slit of width a. The positions of two minima on each side of the central maximum are labeled (drawing not to scale). In other words, a diffraction grating is the tool of choice for separating the colors in incident light. Figure 41: Diffraction grating is an optical device used to disperse light into a spectrum. The diffraction grating, a useful device for analyzing light sources, consists of a large number of equally spaced parallel slits. A transmission grating can be made by cutting parallel lines on a glass plate with a precision ruling machine. The spaces between the lines are transparent to the light, and hence act as separate slits. A plane wave is incident from the left, normal to the plane of the grating. The pattern observed on the screen is the result of the combined effects of interference and diffraction. Each slit produces diffraction, and the diffracted beams interfere with one another to produce the final pattern. Physic 1 Module 3: Optics 20

21 The waves from all slits are in phase as they leave the slits. However, for some arbitrary direction θ measured from the horizontal, the waves must travel different path lengths before reaching a particular point on the viewing screen. The condition for maximum intensity is the same as that for a double slit (see section 3.2.3). However, angular separation of the maxima is generally much greater because the slit spacing is so small for a diffraction grating. The diffraction pattern produced by the grating is therefore described by the equation d sin θ = mλ (69) where m = 0, ± 1, ±2, ±3 and m is the order number; λ is a selected wavelength; d is the spacing of the grooves; and θ is the angle of incidence of light. Equation (69) states the condition for maximum intensity. The diffraction grating is thus an immensely useful tool for the separation of the spectral lines associated with atomic transitions. It acts as a "super prism", separating the different colors of light much more than the dispersion effect in a prism. We can use equation (69) to calculate the wavelength if we know the grating spacing d and the angle θ. If the incident radiation contains several wavelengths, the mthorder maximum for each wavelength occurs at a specific angle. All wavelengths are seen at θ = 0, corresponding to the zeroth-order maximum (m = 0). The first-order maximum (m = 1) is observed at an angle that satisfies the relationship sin θ = λ/d; the second-order maximum (m = 2) is observed at a larger angle θ, and so on. The intensity distribution for a diffraction grating obtained with the use of a monochromatic source is shown in Figure 42. Note the sharpness of the principal maxima and the broadness of the dark areas. This is in contrast to the broad bright fringes characteristic of the two-slit interference pattern (see section 3.2.3). Figure 42: Intensity versus sinθ for a diffraction grating. The zeroth-, first-, and second-order maxima are shown. Diffraction gratings are most useful for measuring wavelengths accurately. Like prisms, diffraction gratings can be used to disperse a spectrum into its wavelength components (see the next section). Of the two devices, the grating is the more precise if one wants to distinguish two closely spaced wavelengths. Example: Light of wavelength 580 nm is incident on a slit having a width of mm. The viewing screen is 2.00 m from the slit. Find the positions of the first dark fringes and the width of the central bright fringe. (Ans. ±3.87 mm, 7.74 mm) Physic 1 Module 3: Optics 21

22 3) Spectroscopy: Dispersion Spectroscope - Spectra Spectroscopy Spectroscopy is the study of the way in which atoms absorb and emit electromagnetic radiation. Spectroscopy pertains to the dispersion of an object's light into its component colors (or energies). By performing the analysis of an object's light, scientists can infer the physical properties of that object (such as temperature, mass, luminosity, and chemical composition). We first realize that light acts like a wave. Light has particle-like properties too (see section 3.5). The wave speed of a light wave is simply the speed of light, and different wavelengths of light manifest themselves as different colors. The energy of a light wave is inversely-proportional to its wavelength; in other words, low-energy light waves have long wavelengths, and highenergy light waves have short wavelengths. Electromagnetic spectrum Physicists classify light waves by their energies (wavelengths). Labeled in increasing energy or decreasing wavelength, we might draw the entire electromagnetic spectrum, as shown in Figure 43. Notice that radio, TV, and microwave signals are all light waves, they simply lie at wavelengths (energies) that our eyes do not respond to. On the other end of the scale, beware the high energy UV, x-ray, and gamma-ray photons. Each one carries a lot of energy compared to their visible-and radio-wave counterparts. Figure 43: The electromagnetic spectrum. Notice how small the visible region of the spectrum is, compared to the entire range of wavelengths. Physic 1 Module 3: Optics 22

23 Dispersion In optics, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency. Media having such a property are termed dispersive media. The most familiar example of dispersion is probably a rainbow, in which dispersion causes the spatial separation of a white light into components of different colors (different wavelengths), see Figure 44. Dispersion is most often described for light waves, but it may occur for any kind of wave that interacts with a medium or passes through an inhomogeneous geometry. In optics, dispersion is sometimes called chromatic dispersion to emphasize its wavelengthdependent nature. The dispersion of light by glass prisms is used to construct spectrometers. Diffraction gratings are also used, as they allow more accurate discrimination of wavelengths. Figure 44: In a prism, material dispersion (a wavelength-dependent refractive index) causes different colors to refract at different angles, splitting white light into a rainbow. The most commonly seen consequence of dispersion in optics is the separation of white light into a color spectrum by a prism. From Snell's law, it can be seen that the angle of refraction of light in a prism depends on the refractive index of the prism material. Since that refractive index varies with wavelength, it follows that the angle that the light is refracted by will also vary with wavelength, causing an angular separation of the colors known as angular dispersion. A white light consists of a collection of component colors. These colors are often observed as white light passes through a triangular prism. Upon passage through the prism, the white light is separated into its component colors - red, orange, yellow, green, blue, and violet. Physic 1 Module 3: Optics 23

24 Figure 45: Diagram of a simple spectroscope. A small slit in the opaque barrier on the left allows a narrow beam of light to pass. The light passes through a prism and is split up into its component colors. The resulting spectrum can be viewed through an eyepiece or simply projected onto a screen. Spectroscope A spectroscope is a device used for splitting a beam of radiation (light) into its component frequencies (or wavelengths) and delivering them to a screen or detector for detailed study (see Figure 45). In other words, spectroscope is an optical system used to observe luminous spectra of light sources. In its most basic form, this device consists of an opaque barrier with a slit in it (to define a beam of light), a prism or a diffraction grating (to split the beam into its component colors), and an eyepiece or screen (to allow the user to view the resulting spectrum). Figure 44 shows such an arrangement. In many large instruments the prism is replaced by a device called a diffraction grating, consisting of a sheet of transparent material with many closely spaced parallel lines ruled on it. The spaces between the lines act as many tiny openings, and light is diffracted as it passes through these openings. Because different wavelengths of electromagnetic radiation are diffracted by different amounts as they pass through a narrow gap, the effect of the grating is to split a beam of light into its component colors. Principle of operation of a spectroscope We use the source of interest to light a narrow slit. A first collimating lens is placed on the path of light to send a parallel beam on a prism or a grating. After the dispersion of light, a second lens projects on a screen the image of the slit, resulting many color lines. Each line correspond to a wavelength. This series of lines constitutes the spectrum of the light source. Examples are shown in Figure 46, including: i. White light is broken up into a continuous spectrum, from red to blue (visible light). ii. A incandescent gas gives bright lines of specific wavelengths; it is an emission spectrum and the position of the lines are characteristic of this gas. Physic 1 Module 3: Optics 24

25 iii. The same cold gas is placed between the source of white light and the spectroscope. It absorbs some of the radiations emitted by this source. Dark lines are observed at the same positions as the bright lines of the previous spectrum. It is an absorption spectrum. SPECTRA In the domain of electromagnetic radiation, a spectrum is a series of radiant energies arranged in order of wavelength or frequency. The entire range of frequencies is subdivided into wide intervals in which the waves have some common characteristic of generation or detection, such as the radio-frequency spectrum, infrared spectrum, visible spectrum, ultraviolet spectrum, and x-ray spectrum. Spectra are also classified according to their origin or mechanism of excitation, as emission, absorption, continuous, line, and band spectra. An emission spectrum is produced whenever the radiation from an excited light source are dispersed. An absorption spectrum is produced against a background of continuous radiation by interposing matter that reduces the intensity of radiation at certain wavelengths or spectral regions. The energies removed from the continuous spectrum by the interposed absorbing medium are precisely those that would be emitted by the medium if properly excited. A continuous spectrum contains an unbroken sequence of wavelengths or frequencies over a long range. Line spectra are discontinuous spectra characteristic of excited atoms and ions, whereas band spectra are characteristic of molecular gases or chemical compounds. Within the visible spectrum, various light wavelengths are perceived as colors ranging from red to blue, depending upon the wavelength of the wave. White light is a combination of all visible colors mixed in equal proportions. This characteristic of light, which enables it to be combined, so that the resultant light is equal to the sum of its constituent wavelengths, is called additive color mixing. The term spectrum (plural form, spectra) is applied to any class of similar entities or properties strictly arrayed in order of increasing or decreasing magnitude. In general, a spectrum is a display or plot of intensity of radiation (particles, photons, or acoustic radiation) as a function of mass, momentum, wavelength, frequency, or some other related quantity. Physic 1 Module 3: Optics 25

26 Figure 46: Examples of continuous spectrum, line spectrum and absorption spectrum Applications of interference and diffraction 1) Applications of interference Interference can be used to measure the wavelength of a monochromatic light (see the example in section 3.2.3). A common example of the applications of interference involves the interference of radio wave signals which occur at the antenna of a home when radio waves from a very distant transmitting station take two different paths from the station to the home. This is relatively common for homes located near mountain cliffs. In such an instance, waves which travel directly from the transmitting station to the antenna interfere with other waves which reflect off the mountain cliffs behind the home and travel back to the antenna, as shown in Figure 47. Physic 1 Module 3: Optics 26

27 Figure 47: An example of radio wave interference. In this case, waves are taking two different paths from the source to the antenna - a direct path and a reflected path. Clearly, each path is represented by a different distance traveled from the source to the home, with the reflected pathway corresponding to the longer distance of the two. If the home is located some distance d from the mountain cliffs, then the waves which take the reflected path to the home will be traveling an extra distance given by the expression 2d. The 2 in this expression is due to the fact that the waves taking the reflected path must travel past the antenna to the cliffs (a distance d) and then back to the antenna from the cliff (a second distance d). Thus, the path difference of 2d results in destructive interference whenever it is equal to a half number of wavelength. Since radio stations transmit their signals at a specific and known frequencies, the wavelengths of these light waves can be determined by relating them to the transmitted frequencies and the light speed in vacuum (3 x 10 8 m/s). Creating holography Holography is a method (technique) of producing a three-dimensional image of an object by recording on a photographic plate or film the pattern of interference formed by a split laser beam and then illuminating the pattern either with a laser or with ordinary light. The technique is widely used as a method for optical image formation and in addition has been successfully used with acoustical (sound) and radio waves. The technique is accomplished by recording the pattern of interference between the wave emanating from the object of interest and a known reference wave, as shown in Figure 48a. In general, the object wave is generated by illuminating the (possibly three-dimensional) subject of interest with a highly coherent beam of light, such as one supplied by a laser source. The waves reflected from the object strike a light-sensitive recording medium, such as photographic film or plate. Simultaneously a portion of the light is allowed to bypass the object and is sent directly to the recording plate, typically by means of a mirror placed next to the object. Thus incident on the recording medium is the sum of the light from the object and a mutually coherent reference wave. Physic 1 Module 3: Optics 27

28 Figure 48b: Obtaining images from a hologram. Figure 48a: Recording a hologram.. The photographic recording obtained is known as a hologram (meaning a total recording ); this record generally bears no resemblance to the original object, but rather is a collection of many fine fringes which appear in rather irregular patterns. Nonetheless, when this photographic transparency is illuminated by coherent light, one of the transmitted wave components is an exact duplication of the original object wave, as shown in Figure 48b. This wave component therefore appears to originate from the object (although the object has long since been removed) and accordingly generates a virtual image of it, which appears to an observer to exist in three-dimensional space behind the transparency. The image is truly threedimensional in the sense that the observer's eyes must refocus to examine foreground and background, and indeed can look behind objects in the foreground simply by moving his or her head laterally. 2) Applications of diffraction Diffraction gratings (see section 3.3.3) Limiting of resolution of an optical instrument The ability of optical instrument such as a microscope to distinguish between closely spaced objects is limited because of the wave nature of light. Consider light waves from different objects far from a narrow slit, and these objects can be considered as two noncoherent point sources S 1 and S 2. If no diffraction occurred, two distinct bright spots (or images) would be observed on the viewing screen. However, because of diffraction, each source is imaged as a bright central region flanked by weaker bright and dark fringes. What is observed on the screen is the sum of two diffraction patterns: one from S 1 and the other from S 2. If the two sources are far enough apart to keep their central maxima from overlapping, their images can be distinguished and are said to be resolved; as a result the observer can see S 1 and S 2 distinguishably. If the sources are close together, however, the two central maxima overlap, and the images are not resolved; as a result the observer can not see S 1 and S 2 distinguishably. Physic 1 Module 3: Optics 28

29 The light diffraction thus imposes a limiting resolution of any optical instrument Duality of lights and particles 1) Photoelectric effect Einstein s photon concept Photoelectric effect Photoelectric effect is a process whereby light falling on a surface knocks electrons out of the surface. The photoelectric effect refers to the emission, or ejection, of electrons from the surface of, generally, a metal in response to incident light, as illustrated in Figure 49. According to Figure 49, when shining a violet light on a clean sodium (Na) metal in a vacuum, electrons were ejected from the surface. It means the photoelectric effect occurred. The remarkable aspects of the photoelectric effect are: Figure 49: Depicting the photoeffect. 1. The electrons are emitted immediately. It means there is no time lag. 2. Increasing the intensity of the light increases the number of photoelectrons ejected, but not their maximum kinetic energy. 3. No electron is emitted until the light has a threshold frequency, no matter how intense the light is. 4. A weak violet light will eject only a few electrons, but their maximum kinetic energies are greater than those for an intense light of longer wavelengths. It means that the maximum kinetic energies of ejected electrons increase when the wavelength of the shining light is shorter. 5. The maximum kinetic energies of the emitted electrons is independent of the intensity of the incident radiation. These observations baffled physicists for many decades, since they cannot be explained if light is thought of only as a wave. If light were to be a wave, both the maximum kinetic energy and the number of the electrons emitted from the metal should increase with an increase in the intensity of light. Observations contradicted this prediction; only the number, and not the maximum kinetic energy, of the electrons increases with the increase of the intensity of the shining light. Einstein s photon concept Einstein (1905) successfully resolved this paradox by proposing that *The incident light consists of individual quanta, called photons, that interact with the electrons in the metal like discrete particles, rather than as continuous waves. Physic 1 Module 3: Optics 29

30 * For a given frequency, or 'color,' of the incident light, each photon carries an energy Physic 1 Module 3: Optics 30 E = hf (70) where h is Planck's constant (h = x Joule seconds) and f the frequency of the light. *Increasing the intensity of the light corresponds, in Einstein's model, to increasing the number of incident photons per unit time (flux), while the energy of each photon remains the same (as long as the frequency of the radiation was held constant). Clearly, in Einstein's model, increasing the intensity of the incident radiation would cause greater numbers of electrons to be ejected, but each electron would carry the same average energy because each incident photon carries the same energy. This assumes that the dominant process consists of individual photons being absorbed by electrons and resulting in the ejection of a single electron for one photon absorbed. Likewise, in Einstein's model, increasing the frequency f, rather than the intensity, of the incident light would increase the maximum kinetic energy of the emitted electrons. Both of these predictions were confirmed experimentally. The photoelectric effect is perhaps the most direct and convincing evidence of the existence of photons and the 'corpuscular' or particle nature of light and electromagnetic radiation. That is, it provides undeniable evidence of the quantization of the electromagnetic field and the limitations of the classical field equations of Maxwell. Albert Einstein received the Nobel prize in physics in 1921 for explaining the photoelectric effect and for his contributions to theoretical physics. Energy contained within the incident light is absorbed by electrons within the metal, giving the electrons sufficient energy to be knocked out of, that is, emitted from, the surface of the metal. According to the classical Maxwell wave theory of light, the more intense the incident light is the greater the energy with which the electrons should be ejected from the metal. That is, the maximum kinetic energy of ejected (photoelectric) electrons should increase with the intensity of the incident light. This is, however, not the case. The minimum energy required to eject an electron from the surface of a metal is called the photoelectric work function of the metal, often denoted as φ. Thus the condition for the photoeffect to occur is hf φ (71) Let φ = hf 0 (72) The condition for the photoeffect to occur becomes f o is called the threshold frequency of the metal. f f 0 (73)

31 Using f = c/λ and letting f 0 = c/λ 0, equation (73) becomes λ λ 0 (74) λ 0 is called the threshold wavelength of the metal. φ, f 0, and λ 0 depend on the nature of the metal of interest. (71), (73), and (74) set the condition for the photoelectric effect to occur. The maximum kinetic energy of the emitted electrons, E kinmax, is thus given by the energy of the photon minus the photoelectric work function E kinmax = hf - φ (75) E kinmax thus depends on the frequency of light falling on the surface, but not on the intensity of the shining light. From equation (75) we see that the emitted electrons move with greater speed if the applied light has a higher frequency provided that (71) is satisfied. Example: Lithium, beryllium, and mercury have work functions of 2.3 ev, 3.90 ev, and 4.50 ev, respectively. If 400-nm light is incident on each of these metals, determine (a) which metal exhibits the photoelectric effect and (b) the maximum kinetic energy of the emitted electrons in each case. (Ans. (b) 0.81 ev) 2) Electromagnetic waves and photons Light as a wave In the early days of physics (say, before the nineteenth century), very little was known about the nature of light, and one of the great debates about light was over the question of whether light is made of a bunch of "light particles," or whether light is a wave. Around 1800, a man named Thomas Young apparently settled the question by performing an experiment in which he shone light through very narrow slits and observed the result (see section 3.2.3). Here's the idea behind it. Suppose you have a whole bunch of ping-pong balls. You stand back about fifteen feet from a doorway, and one by one you dip the balls in paint and throw them through a door, at a wall about 5 feet behind the door. You will get a bunch of colored dots on the wall, scattered throughout an area the same shape as the door you are throwing them through. This is how particles (such as ping-pong balls) behave. Physic 1 Module 3: Optics 31 On the other hand, waves do not behave this way. Think of water waves. When a wave encounters an obstacle, it goes around it and closes in behind it. When a wave passes through an opening, it

32 spreads out when it reaches the other side (diffraction, see section 3.3.1). And under the right conditions, a wave passing through an opening can form interesting diffraction patterns on the other side, which can be deduced mathematically. Young shone momochromatic light through two very narrow slits, very close together. He then observed the result on a screen. Now if light is made up of particles, then the particles should pass straight through the slits and produce two light stripes on the screen, approximately the same size as the slits. (Just like the ping-pong balls in the picture above.). On the other hand, if light is a wave, then the two waves emerging from the two slits will interfere with each other and produce a pattern of many stripes, not just two. Young found the interference pattern with many stripes, indicating that light is a wave. Later in the nineteenth century, James Clerk Maxwell determined that light is an electromagnetic wave: a transverse wave of oscillating electric and magnetic fields. When Heinrich Hertz experimentally confirmed Maxwell's result, the struggle to understand light was finished. Light as particles As mentioned earlier, when light is shone on a metal surface, electrons can be ejected from that surface. This is called the photoelectric effect. Without going into detail, if one assumes that light is a wave, as Young showed, then there are certain features of the photoelectric effect that simply seem impossible. What Einstein showed is that if one assumes that light is made up of particles (now called "photons"), the photoelectric effect can be explained successfully, as discussed in the previous section. 3) Wave-particle duality - De Broglie s postulate Wave-Particle Duality Is light a wave, or is light a flow of particles? Under certain conditions, such as when we shine it through narrow slits and look at the result, it behaves as only a wave can. Under other conditions, such as when we shine it on a metal and examine the electrons that comes off, light behaves as only particles can. This multiple personality of light is referred to as wave-particle duality. Physic 1 Module 3: Optics 32

33 Figure 50: Left: photoeffect showing particle nature of light; Right: Davisson-Germer experiment showing wave nature of electrons. Light behaves as a wave, or as particles, depending on what we do with it, and what we try to observe. A wave-particle dual nature was soon found to be characteristic of electrons as well. The evidence for the description of light as waves was well established before the time when the photoelectric effect first introduced firm evidence of the particle nature of light. On the other hand, the particle properties of electrons was well documented when the De Broglie s postulate and the subsequent experiment by Davisson and Germer established the wave nature of electrons, as shown in Figure 50. De Broglie s postulate In 1924 Louis de Broglie proposed the idea that all matter displays the wave-particle duality as photons do. According to De Broglie s postulate, for all matter and for electromagnetic radiation alike, the energy E of the particle is related to the frequency f of its associated wave, by the Planck relation. E = hf (76) and that the momentum p of the particle is related to its wavelength λ by what is known as the De Broglie relation. p = h λ (77) where h is Planck's constant. The Davisson Germer experiment was a physics experiment conducted in 1927 which confirmed De Broglie s hypothesis, which says that particles of matter (such as electrons) have wave properties. This is a demonstration of wave-particle duality of electrons. Description of the Davisson Germer experiment The experiment consisted of firing an electron beam from an electron gun on a nickel crystal at normal incidence (i.e., perpendicular to the surface of the crystal), as shown in Figure 50. The angular dependence of the reflected electron intensity was measured by an electron Physic 1 Module 3: Optics 33