Module 3 optics 2023 (1)

MINISTRY OF EDUCATION AND TRAINING NONG LAM UNIVERSITY FACULTY OF CHEMICAL ENGINEERING AND FOOD TECHNOLOGY Course: Physics Module 3: Optics and wave phenomena Instructor: Dr Nguyen Thanh Son Academic year: 2021-2022 Contents Module 3: Optics and wave phenomena 3.1 Wave review 3.1.1 Description of a wave 3.1.2 Transverse waves and longitudinal waves 3.1.3 Mathematical description of a traveling (propagating) wave with constant amplitude 3.1.4 Electromagnetic waves 3.1.5 Spherical and plane waves 3.2 Interference of sound waves and light waves 3.2.1 Interference of sinusoidal waves – Coherent sources 3.2.2 Interference of sound waves 3.2.3 Interference of light waves 3.3 Diffraction and spectroscopy 3.3.1 Introduction to diffraction 3.3.2 Diffraction by a single narrow slit - Diffraction gratings 3.3.3 Spectroscopy: Dispersion – Spectroscope – Spectra 3.4 Applications of interference and diffraction 3.4.1 Applications of interference 3.4.2 Applications of diffraction 3.5 Wave-particle duality of matter 3.5.1 Photoelectric effect – Einstein’s photon concept 3.5.2 Electromagnetic waves and photons 3.5.3 Wave-particle duality – De Broglie’s postulate Physic Module 3: Optics and waves 3.1 Wave review 3.1.1 Description of a propagating wave Figure 23 Representation of a typical wave, showing its direction of motion (direction of travel), wavelength, crests, troughs and amplitude • 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 • In wave motion, 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 • 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 While 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 Module 3: Optics and waves 3.1.2 Transverse waves and longitudinal waves In terms of the direction of particles’ 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’ 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’ vibrations along the wave's Figure 24 When an object bobs up and down on a direction of travel; examples include sound ripple in a pond, it experiences an elliptical waves in the air 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 3.1.3 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 = u Max 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 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 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 • 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 Physic Module 3: Optics and waves f= 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 (one 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, vp or v, which gives the rate at which the wave propagates, is given by v= ω k (50) The second is the group velocity, vg, which gives the velocity at which variations in the shape of the wave pattern propagate through space This is also the rate at which information can be transmitted by the wave It is given by vg = ∂ω ∂k (51) • The wavelength (denoted as λ) is the distance between two successive crests (or troughs) of a wave, as shown in Figure 23 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 MHz • The wave number 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 = Hz, λ = 1.25 m, v = 2.5 m/s) Physic Module 3: Optics and waves Example: A sinusoidal wave traveling in the positive x-direction has an amplitude of 15.0 cm, a wavelength of 40.0 cm, and a frequency of 8.00 Hz Find the angular wave number k, period T, angular frequency ω, and speed v of the wave Ans k = 0.157 rad/cm; T = 0.125 s; ω = 50.3 rad/s; v = 3.2 m/s 3.1.4 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 are 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 electromagnetic waves, as shown in Figure 25 In vacuum, light propagates with phase speed: v = c = x 108 m/s Physic Module 3: Optics and waves • 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 3.1.5 Spherical and plane waves • If a small spherical body, considered 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 to 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 radial 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 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 along the x axis, which means that the 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) Physic Module 3: Optics and waves 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 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 has wave fronts that are planes parallel to each other, rather than spheres of increasing radius (Figure 28) 3.2 Interference of sound waves and light waves ♦ Interference of waves • What happens when two waves meet while they travel through the same medium? What effect 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 a phenomenon in which two or more waves to reinforce or partially cancel each other 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 opposite directions in the same medium Suppose that each is displaced upward 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 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 Physic Module 3: Optics and waves 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 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 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 directions 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 diagrams 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 Figure 31 Depicting the snapshots of the medium for the instant of complete overlap, two pulses of the same amplitude (one upward and one there is no resulting downward) before and during interference; displacement of the particles of the interference is destructive the medium 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 Destructive interference leads to only a momentary condition in which the medium's displacement is less than the displacement of the larger-amplitude wave The two interfering waves not need to have equal amplitudes in Figure 32 Depicting the before and during opposite directions for destructive interference snapshots of the medium for two pulses of interference to occur For example, a different amplitudes (one upward, +1 unit and one pulse with a maximum displacement of downward, -2 unit); the interference is destructive +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 • 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 meet, the resulting displacement of the medium at any location is the algebraic sum of the displacements of the individual waves at that location Physic Module 3: Optics and waves • In the cases mentioned above, the summing of the individual displacements for locations of complete overlap was easy and given in the below table Maximum displacement of Pulse +1 −1 +1 +1 Maximum displacement of Pulse +1 −1 −1 −2 Maximum resulting displacement +2 −2 −1 3.2.1 Interference of sinusoidal waves – Coherent sources ♦ Mathematics of two-point source interference • 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 the larger amplitude of the individual waves Light waves also interfere with each other Fundamentally, 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 of waves have the same frequency • The sources of waves must maintain a constant phase with respect to each other 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 of the same frequency (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 sources of waves - 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 waves emerging from the two slits are coherent because a single source produces the original light beam and the two slits only serve 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 O1 and O2 The waves meet at point P, and according to the principle of superposition, the resultant vibration at P is given by uP = u1 + u2 = Asin(kx1 − ωt) + Asin(kx2 − ωt) Physic Module 3: Optics and waves 10 (55) interference and diffraction Each slit produces diffraction, and the diffracted beams interfere with one another to produce the final pattern • 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) 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 dsin θmax/m = 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 (grating spacing) 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 Figure 42 Intensity versus sin θ for a diffraction grating The zeroth-, first-, and second-order maxima are shown • 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 mth-order 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 θmax/1 = 1λ/d; the second-order maximum (m = 2) is observed at a larger angle θ, and so on (Figure 42) • 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) • 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: The wavelengths of the hydrogen alpha line and the hydrogen beta line are 653.4 nm and 580.8 nm, respectively; using a grating with 2.00 x 10 lines (grooves/slits) per meter, what is the angular separation for these two spectral lines in the first order? Physic Module 3: Optics and waves 21 Solution In this problem, d = divided by the number of lines per meter = 1/(2.00 x 105/m) = x 10-6 m For λ1 = 653.4 x 10 -9 m and m = 1, using (69) sin θ1 = (1)(653.4 x 10-9 m)/(5 x 10-6 m) = o θ1 = 7.51 0.131 θ2 = For λ2 = 580.8 x 10 -9 m and m = sin θ2 = (1)(580.8 x 10 -9 m)/(5 x 10-6 m) = 0.116 o 6.67 As a result, in the first order, the angular separation for the α and β lines is (7.51 - 6.67)o = 0.84 o 3.3.3 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 • The 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 or wavelengths Labeling 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 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 Physic Module 3: Optics and waves 22 Figure 43 The electromagnetic spectrum Notice how small the visible region of the spectrum is, compared to the entire range of wavelengths ♦ 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 Physic Module 3: Optics and waves 23 • A white light consists of a collection of component colors These colors are often observed as white light passes through a triangular prism Upon passing through the prism, the white light is separated into its component colors - red, orange, yellow, green, blue, and violet 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 onto 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 45 shows such an arrangement • In many large instruments, the prism is replaced by 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 collimating lens is placed on the path of light to send a parallel beam on a prism or a diffraction grating After the dispersion of the light, a second lens projects on a screen the image of the slit, resulting many color lines Each line corresponds to a wavelength This series of lines constitutes the spectrum of the light source Examples are shown in Figure 46, including: Physic Module 3: Optics and waves 24 i White light is broken up into a continuous spectrum, from red to blue (visible light) ii An incandescent gas gives bright lines of specific wavelengths; it is an emission spectrum and the positions of the lines are characteristic of the gas 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 • 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 • 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 is dispersed - 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 - 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 • 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 Physic Module 3: Optics and waves 25 Figure 46 Examples of continuous spectrum, line spectrum, and absorption spectrum 3.4 Applications of interference and diffraction 3.4.1 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 Module 3: Optics and waves 26 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 at 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 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 odd multiple of half wavelengths Since radio stations transmit their signals at 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 108 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 wave from the object and a mutually coherent reference wave Physic Module 3: Optics and waves 27 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 3.4.2 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 two noncoherent point sources S1 and S2 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 S1 and the other from S2 • 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 S1 and S2 distinguishably • If the sources are close together, however, the two central maxima overlap, and the images are not resolved; as a result the observer cannot see S1 and S2 distinguishably Physic Module 3: Optics and waves 28 • The light diffraction thus imposes a limiting resolution of any optical instrument 3.5 Duality of light and particle 3.5.1 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 by 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 Figure 49 Depicting the photoelectric effect • The remarkable aspects of the photoelectric effect are: The electrons are emitted immediately It means there is no time lag Increasing the intensity of the light increases the number of photoelectrons ejected, but not their maximum kinetic energy No electron is emitted until the light has a threshold frequency, no matter how intense the light is 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 The maximum kinetic energy 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 * For a given frequency, or 'color,' of the incident light, each photon carries an energy E = hf Physic Module 3: Optics and waves 29 (70) where h is Planck's constant (h = 6.626069 x 10-34 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 photoelectric effect to occur is Let hf ≥ φ (71) φ = hf0 (72) The condition for the photoelectric effect to occur becomes f ≥ f0 (73) fo is called the threshold frequency of the metal Using f = c/λ and letting f0 = c/λ0, Equation 73 becomes λ ≤ λ0 Physic Module 3: Optics and waves 30 (74) λ0 is called the threshold wavelength of the metal φ, f0, and λ0 depend on the nature of the metal of interest • Equations 71, 73, and 74 set the condition for the photoelectric effect to occur • The maximum kinetic energy of the emitted electrons, EkinMax, is thus given by the energy of the photon minus the photoelectric work function EkinMax = hf − φ (75) EkinMax thus depends on the frequency of the 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 Equation 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) 3.5.2 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 (table tennis) 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 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 • On the other hand, waves 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 spreads out when it reaches the other side (diffraction, see Section 3.3.1) And under the right Physic Module 3: Optics and waves 31 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 3.5.3 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 come off, light behaves as only particles can This multiple personality of light is referred to as wave-particle duality Physic Module 3: Optics and waves 32 Figure 50 Left: photoelectric effect 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 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 were 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 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’s 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 Module 3: Optics and waves 33 detector and was determined to have the diffraction patterns (similar as those predicted by Bragg for x-rays) nλ = d sin θ (78) where • • • • n is an integer determined by the order given, λ is the wavelength of x-rays, moving electrons, protons, and neutrons, d is the spacing between the planes in the atomic lattice, and θ is the angle between the incident ray and the scattering planes Before the acceptance of De Broglie’s hypothesis, diffraction was a property that was thought to be only exhibited by waves Therefore, the presence of any diffraction effects by matter demonstrated the wave-like nature of matter When De Broglie wavelength was inserted into Equation 78, the observed diffraction pattern was found as predicted, thereby experimentally confirming De Broglie’s hypothesis for electrons Physic Module 3: Optics and waves 34 REFERENCES 1) Halliday, David; Resnick, Robert; Walker, Jearl (1999), Fundamentals of Physics, 7th ed., John Wiley & Sons, Inc 2) Feynman, Richard; Leighton, Robert; Sands, Matthew (1989), Feynman Lectures on Physics, Addison-Wesley 3) Serway, Raymond; Faughn, Jerry (2003), College Physics, 7th ed., Thompson, Brooks/Cole 4) Sears, Francis; Zemansky Mark; Young, Hugh (1991), College Physics, 7th ed., AddisonWesley 5) Beiser, Arthur (1992), Physics, 5th ed., Addison-Wesley Publishing Company 6) Jones, Edwin; Childers, Richard (1992), Contemporary College Physics, 7th ed., AddisonWesley 7) Alonso, Marcelo; Finn, Edward (1972), Physics, 7th ed., Addison-Wesley Publishing Company 8) Michels, Walter; Correll, Malcom; Patterson, A L (1968), Foundations of Physics, 7th ed., Addison-Wesley Publishing Company 9) Hecht, Eugene (1987), Optics, 2th ed., Addison-Wesley Publishing Company 10) Eisberg, R M (1961), Modern Physics, John Wiley & Sons, Inc 11) Reitz, John; Milford, Frederick; Christy Robert (1993), Foundations of Electromagnetic Theory, 4th ed., Addison-Wesley Publishing Company 12) Priest, Joseph (1991), Energy: Principle, Problems, Alternatives, 4th ed., Addison-Wesley Publishing Company 13) Giambattista Alan; 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