Optics and Photonics PDF

# The Nature and Propagation of Light ## **Learning Goals** - What light rays are, and how they are related to wave fronts. - The laws that govern the reflection and refraction of light. - The circumstances under which light is totally reflected at an interface. - The consequences of the speed of l...

# The Nature and Propagation of Light ## **Learning Goals** - What light rays are, and how they are related to wave fronts. - The laws that govern the reflection and refraction of light. - The circumstances under which light is totally reflected at an interface. - The consequences of the speed of light in a material being different for different wavelengths. - How to make polarized light out of ordinary light. - How the scattering of light explains the blue color of the sky. - How Huygens's principle helps us analyze reflection and refraction. ## 33.1 **The Nature of Light** - Until the time of Isaac Newton (1642-1727), most scientists thought that light consisted of streams of particles (called corpuscles) emitted by light sources. - Galileo and others tried (unsuccessfully) to measure the speed of light. - Around 1665, evidence of wave properties of light began to be discovered. - By the early 19th century, evidence that light is a wave had grown very persuasive. - In 1873, James Clerk Maxwell predicted the existence of electromagnetic waves and calculated their speed of propagation. - This development, along with the experimental work of Heinrich Hertz starting in 1887, showed conclusively that light is indeed an electromagnetic wave. ## **The Two Personalities of Light** - The wave picture of light is not the whole story, however. - Several effects associated with emission and absorption of light reveal a particle aspect, in that the energy carried by light waves is packaged in discrete bundles called photons or quanta. - These apparently contradictory wave and particle properties have been reconciled since 1930 with the development of quantum electrodynamics, a comprehensive theory that includes both wave and particle properties. - The propagation of light is best described by a wave model, but understanding emission and absorption requires a particle approach. - The fundamental sources of all electromagnetic radiation are electric charges in accelerated motion. - All bodies emit electromagnetic radiation as a result of thermal motion of their molecules; this radiation, called thermal radiation, is a mixture of different wavelengths. - At sufficiently high temperatures, all matter emits enough visible light to be self-luminous; a very hot body appears "red-hot" or "white-hot." - Thus hot matter in any form is a light source. - Familiar examples are a candle flame, hot coals in a campfire, and the coils in an electric toaster oven or room heater. - Light is also produced during electrical discharges through ionized gases. - The bluish light of mercury-arc lamps, the orange-yellow of sodium-vapor lamps, and the various colors of "neon" signs are familiar. - A variation of the mercury-arc lamp is the fluorescent lamp. - This light source uses a material called a phosphor to convert the ultraviolet radiation from a mercury arc into visible light. - This conversion makes fluorescent lamps more efficient than incandescent lamps in transforming electrical energy into light. - In most light sources, light is emitted independently by different atoms within the source; in a laser, by contrast, atoms are induced to emit light in a cooperative, coherent fashion. - The result is a very narrow beam of radiation that can be enormously intense and that is much more nearly monochromatic, or single-frequency, than light from any other source. - Lasers are used by physicians for microsurgery, in a DVD or Blu-ray player to scan the information recorded on a video disc, in industry to cut through steel and to fuse high-melting-point materials, and in many other applications. - No matter what its source, electromagnetic radiation travels in vacuum at the same speed *c*. - As we saw in Sections 1.3 and 32.1, this speed is defined to be $c = 2.99792458 \times 10^8 m/s$ or $3.00 \times 10^8 m/s$ to three significant figures. - The duration of one second is defined by the cesium clock, so one meter is defined to be the distance that light travels in 1/299,792,458 s. ## **Waves, Wave Fronts, and Rays** - We often use the concept of a wave front to describe wave propagation. - We introduced this concept in Section 32.2 to describe the leading edge of a wave. - More generally, we define a wave front as the locus of all adjacent points at which the phase of vibration of a physical quantity associated with the wave is the same. - That is, at any instant, all points on a wave front are at the same part of the cycle of their variation. - When we drop a pebble into a calm pool, the expanding circles formed by the wave crests, as well as the circles formed by the wave troughs between them, are wave fronts. - Similarly, when sound waves spread out in still air from a pointlike source, or when electromagnetic radiation spreads out from a pointlike emitter, any spherical surface that is concentric with the source is a wave front. - In diagrams of wave motion we usually draw only parts of a few wave fronts, often choosing consecutive wave fronts that have the same phase and thus are one wavelength apart, such as crests of water waves. - Similarly, a diagram for sound waves might show only the "pressure crests," the surfaces over which the pressure is maximum, and a diagram for electromagnetic waves might show only the "crests" on which the electric or magnetic field is maximum. - We will often use diagrams that show the shapes of the wave fronts or their cross sections in some reference plane. - For example, when electromagnetic waves are radiated by a small light source, we can represent the wave fronts as spherical surfaces concentric with the source or, as in Fig. 33.4a, by the circular intersections of these surfaces with the plane of the diagram. - Far away from the source, where the radii of the spheres have become very large, a section of a spherical surface can be considered as a plane, and we have a plane wave like those discussed in Sections 32.2 and 32.3. - To describe the directions in which light propagates, it's often convenient to represent a light wave by rays rather than by wave fronts. - In a particle theory of light, rays are the paths of the particles. - From the wave viewpoint a ray is an imaginary line along the direction of travel of the wave. - In Fig. 33.4a the rays are the radii of the spherical wave fronts, and in Fig. 33.4b they are straight lines perpendicular to the wave fronts. - When waves travel in a homogeneous isotropic material the rays are always straight lines normal to the wave fronts. - At a boundary surface between two materials, such as the surface of a glass plate in air, the wave speed and the direction of a ray may change, but the ray segments in the air and in the glass are straight lines. - The next several chapters will give you many opportunities to see the interplay of the ray, wave, and particle descriptions of light. - The branch of optics for which the ray description is adequate is called geometric optics; the branch dealing specifically with wave behavior is called physical optics. - This chapter and the following one are concerned mostly with geometric optics. - In Chapters 35 and 36 we will study wave phenomena and physical optics. ## 33.2 **Reflection and Refraction** - In this section we'll use the ray model of light to explore two of the most important aspects of light propagation: reflection and refraction. - When a light wave strikes a smooth interface separating two transparent materials, the wave is in general partly reflected and partly refracted (transmitted) into the second material. - For example, when you look into a restaurant window from the street, you see a reflection of the street scene, but a person inside the restaurant can look out through the window at the same scene as light reaches him by refraction. - The segments of plane waves shown in Fig. 33.5a can be represented by bundles of rays forming beams of light. - For simplicity we often draw only one ray in each beam. - Representing these waves in terms of rays is the basis of geometric optics. - We begin our study with the behavior of an individual ray. - We describe the directions of the incident, reflected, and refracted (transmitted) rays at a smooth interface between two optical materials in terms of the angles they make with the normal (perpendicular) to the surface at the point of incidence. - If the interface is rough, both the transmitted light and the reflected light are scattered in various directions, and there is no single angle of transmission or reflection. - Reflection at a definite angle from a very smooth surface is called specular reflection; scattered reflection from a rough surface is called diffuse reflection. - Both kinds of reflection can occur with either transparent materials or opaque materials that do not transmit light. - The vast majority of objects in your environment (including plants, other people, and this book) are visible to you because they reflect light in a diffuse manner from their surfaces. - Our primary concern, however, will be with specular reflection from a very smooth surface such as highly polished glass or metal. - Unless stated otherwise, when referring to "reflection" we will always mean specular reflection. - The index of refraction of an optical material (also called the refractive index), denoted by n, plays a central role in geometric optics. $n = \frac{c}{v}$ - Light always travels more slowly in a material than in vacuum, so the value of v in anything other than vacuum is always greater than unity. - For vacuum, n = 1. - Since n is a ratio of two speeds, it is a pure number without units. - We described the relationship of the value of n to the electric and magnetic properties of a material. ## **The Laws of Reflection and Refraction** - Experimental studies of reflection and refraction at a smooth interface between two optical materials lead to the following conclusions. - The incident, reflected, and refracted rays and the normal to the surface all lie in the same plane. - This plane, called the plane of incidence, is perpendicular to the plane of the boundary surface between the two materials. - We always draw ray diagrams so that the incident, reflected, and refracted rays are in the plane of the diagram. - The angle of reflection (θ) is equal to the angle of incidence (θ) for all wavelengths and for any pair of materials. That is, in Fig. 33.5c, $\theta_r = \theta_i$ - This relationship, together with the observation that the incident and reflected rays and the normal all lie in the same plane, is called the law of reflection. - For monochromatic light and for a given pair of materials, a and b, on opposite sides of the interface, the ratio of the sines of the angles θi and θr, where both angles are measured from the normal to the surface, is equal to the inverse ratio of the two indexes of refraction. $\frac{sin\theta_i}{sin\theta_r} = \frac{n_b}{n_a}$ - This result, together with the observation that the incident and refracted rays and the normal all lie in the same plane, is called the law of refraction or Snell's law, after the Dutch scientist Willebrord Snell. - While these results were first observed experimentally they can be derived theoretically from a wave description of light. - We do this in Section 33.7. - Equations show that when a ray passes from one material (a) into another material (b) having a larger index of refraction, (n > n₁) and hence a slower wave speed, the angle θ with the normal is smaller in the second material than the angle θi in the first; hence the ray is bent toward the normal. - When the second material has a smaller index of refraction than the first material, (n < n₁) and hence a faster wave speed, the ray is bent away from the normal. - No matter what the materials on either side of the interface, in the case of normal incidence the transmitted ray is not bent at all. - In this case θi = 0 and sine = 0, so (θ) is also equal to zero; the transmitted ray is also normal to the interface. - From Eq.(33.2), θr is also equal to zero, so the reflected ray travels back alongt the same path as the incident ray. - The law of refraction explains why a partially submerged ruler or drinking straw appears bent; light rays coming from below the surface changes in direction at the air-water interface, so the rays appear to be coming from a position above their actual point of origin. ## **Applications of Total Internal Reflection** - Total internal reflection finds numerous uses in optical technology. - As an example, consider glass with index of refraction n = 1.52. - If light propagating within this glass encounters a glass-air interface, the critical angle is $sin \theta_{crit} = \frac{1}{1.52}$ $ \theta_{crit} = 41.1^o$ - The light will be totally reflected if it strikes the glass-air surface at an angle of 41.1^o or larger. - Because the critical angle is slightly smaller than 45°, it is possible to use a prism with angles of 45°-45°-90° as a totally reflecting surface. - As reflectors, totally reflecting prisms have some advantages over metallic surfaces such as ordinary coated-glass mirrors. - While no metallic surface reflects 100% of the light incident on it, light can be totally reflected by a prism. - These reflecting properties of a prism are unaffected by tarnishing. - A 45°-45°-90° prism, used as in Fig. 33.14a, is called a Porro prism. - Light enters and leaves at right angles to the hypotenuse and is totally reflected at each of the shorter faces. - The total change of direction of the rays is 180°. - Binoculars often use combinations of two Porro prisms. ## 33.3 **Total Internal Reflection** - We have described how light is partially reflected and partially transmitted at an interface between two materials with different indexes of refraction. - Under certain circumstances, however, all of the light can be reflected back from the interface, with none of it being transmitted, even though the second material is transparent. - Several rays are shown radiating from a point source in material *a* with index of refraction n*a*. - The rays strike the surface of a second material *b* with index n*b*, where n*a* > n*b*. - Materials *a* and *b* could be water and air, respectively. - From Snell's law of refraction, $n_a sin \theta_i = n_b sin \theta_r $ - Because n*a*/n*b* is greater than unity, sin θ*r* is larger than sin θ*i*; the ray is bent away from the normal. - Thus there must be some value of θ*i* less than 90° for which sin θ*r* = 1 and θ*r* = 90°. - This is shown by ray 3 in the diagram, which emerges just grazing the surface at an angle of refraction of 90°. - The angle of incidence for which the refracted ray emerges tangent to the surface is called the critical angle, denoted by θ*crit*. - A more detailed analysis using Maxwell's equations shows that as the incident angle approaches the critical angle, the transmitted intensity approaches zero. - If the angle of incidence is larger than the critical angle, sin θ*r* would have to be greater than unity, which is impossible. - Beyond the critical angle, the ray cannot pass into the upper material; it is trapped in the lower material and is completely reflected at the boundary surface. - This situation, called total internal reflection, occurs only when a ray in material *a* is incident on a second material *b* whose index of refraction is smaller than that of material *a* (that is, n*a* > n*b*). - We can find the critical angle for two given materials *a* and *b* by setting θ*r* = 90° (sin θ*r* = 1) in Snell's law. - We then have $ sin \theta_{crit} = \frac{n_b}{n_a}$. - Total internal reflection will occur if the angle of incidence is larger than or equal to θ*crit*. ## 33.4 **Dispersion** - Ordinary white light is a superposition of waves with all visible wavelengths. - The speed of light in vacuum is the same for all wavelengths, but the speed in a material substance is different for different wavelengths. - Therefore the index of refraction of a material depends on wavelength. - The dependence of wave speed and index of refraction on wavelength is called dispersion. - Figure 33.17 shows the variation of index of refraction *n* with wavelength for some common optical materials. - Note that the horizontal axis of this figure is the wavelength λo of the light in vacuum; the wavelength in the material is given by λ = λo/n. - In most materials the value of n decreases with increasing wavelength and decreasing frequency, and thus n increases with decreasing wavelength and increasing frequency. - In such a material, light of longer wavelength has greater speed than light of shorter wavelength. - Figure 33.18 shows a ray of white light incident on a prism. - The deviation (change of direction) produced by the prism increases with increasing index of refraction and frequency and decreasing wavelength. - So violet light is deviated most, and red is deviated least. - When it comes out of the prism, the light is spread out into a fan-shaped beam. - The light is said to be dispersed into a spectrum. - The amount of dispersion depends on the difference between the refractive indexes for violet light and for red light. - From Fig. 33.17 we can see that for fluorite, the difference between the indexes for red and violet is small, and the dispersion will also be small. - A better choice of material for a prism whose purpose is to produce a spectrum would be silicate flint glass, for which there is a larger difference in the value of n between red and violet. - As we mentioned in Section 33.3, the brilliance of diamond is due in part to its unusually large refractive index; another important factor is its large dispersion, which causes white light entering a diamond to emerge as a multicolored spectrum. - Crystals of rutile and of strontium titanate, which can be produced synthetically, have about eight times the dispersion of diamond. - When a wave passes from one material into a second material with larger index of refraction, so n*a*< n*b*, the wave speed decreases. - The wavelength λ = λo/n in the second material is then shorter than the wavelength λo/n in the first material. - If instead the second material has a smaller index of refraction than the first material, so n*a* > n*b*, then the wave speed increases. Then the wavelength λ in the second material is longer than the wavelength λo in the first material. - This makes intuitive sense; the waves get "squeezed" (the wavelength gets shorter) if the wave speed decreases and get "stretched" (the wavelength gets longer) if the wave speed increases.. ## 33.5 **Polarization** - Polarization is a characteristic of all transverse waves. - For a string that in equilibrium lies along the x-axis, the displacements may be along the y-direction. - In this case the string always lies in the xy-plane. - But the displacements might instead be along the z-axis; then the string always lies in the xz-plane. - When a wave has only y-displacements, we say that it is linearly polarized in the y-direction; a wave with only z-displacements is linearly polarized in the z-direction. - For mechanical waves we can build a polarizing filter, or polarizer, that permits only waves with a certain polarization direction to pass. - This same language can be applied to electromagnetic waves, which also have polarization. - As we learned in Chapter 32, an electromagnetic wave is a transverse wave; the fluctuating electric and magnetic fields are perpendicular to each other and to the direction of propagation. - We always define the direction of polarization of an electromagnetic wave to be the direction of the electric-field vector *E*, not the magnetic field, because many common electromagnetic-wave detectors respond to the electric forces on electrons in materials, not the magnetic forces. ## **Polarizing Filters** - Waves emitted by a radio transmitter are usually linearly polarized. - The vertical antennas that are used for radio broadcasting emit waves that, in a horizontal pane around the antenna, are polarized in the vertical direction (parallel to the antenna). - The situation is different for visible light. - Light from incandescent light bulbs and fluorescent light fixtures is not polarized. - The "antennas" that radiate light waves are the molecules that make up the sources. - The waves emitted by any one molecule may be linearly polarized, like those from a radio antenna. - But any actual light contains a tremendous number of molecules with random orientations, so the emitted light is a random mixture of waves that are linearly polarized in all possible transverse directions. - Such light is called unpolarized light or natural light. - To create polarized light from unpolarized natural light requires a filter that is analogous to the slot for mechanical waves. - Polarizing filters for electromagnetic waves have different details of construction, depending on the wavelength. - For microwaves with a wavelength of a few centimeters, a good polarizer is an array of closely spaced, parallel conducting wires that are insulated from each other. - Electrons are free to move along the length of the conducting wires and will do so in response to a wave whose E field is parallel to the wires. - The resulting currents in the wires dissipate energy by I2R heating; the dissipated energy comes from the wave, so whatever wave passes through the grid is greatly reduced in amplitude. - Waves with E oriented perpendicular to the wires pass through almost unaffected, since electrons cannot move through the air between the wires. - Hence a wave that passes through such a filter will be predominantly polarized in the direction perpendicular to the wires. - The most common polarizing filter for visible light is a material known by the trade name Polaroid, widely used for sunglasses and polarizing filters for camera lenses. - This material incorporates substances that have dichroism, a selective absorption in which one of the polarized components is absorbed much more strongly than the other. - A Polaroid filter transmits 80% or more of the intensity of a wave that is polarized parallel to the polarizing axis of the material, but only 1% or less for waves that are polarized perpendicular to this axis. - In one type of Polaroid filter, long-chain molecules within the filter are oriented with their axis perpendicular to the polarizing axis; these molecules preferentially absorb light that is polarized along their length, much like the conducting wires in a polarizing filter for microwaves. ## **Using Polarizing Filters** - An ideal polarizing filter (polarizer) passes 100% of the incident light that is polarized parallel to the filter's polarizing axis but completely blocks all light that is polarized perpendicular to this axis. - Such a device is an unattainable idealization, but the concept is useful in clarifying the basic ideas. - In the following discussion we will assume that all polarizing filters are ideal. - In Fig. 33.23 unpolarized light is incident on a flat polarizing filter. - The E vector of the incident wave can be represented in terms of components parallel and perpendicular to the polarizer axis; only the component of E parallel to the polarizing axis is transmitted. - Hence the light emerging from the polarizer is linearly polarized parallel to the polarizing axis. - When unpolarized light is incident on an ideal polarizer as in Fig. 33.23, the intensity of the transmitted light is exactly half that of the incident unpolarized light, no matter how the polarizing axis is oriented. - Here's why: We can resolve the E field of the incident wave into a component parallel to the polarizing axis and a component perpendicular to it. - Because the incident light is a random mixture of all states of polarization, these two components are, on average, equal. - The ideal polarizer transmits only the component that is parallel to the polarizing axis, so half the incident intensity is transmitted. - What happens when the linearly polarized light emerging from a polarizer passes through a second polarizer, or analyzer, as in Fig. 33.24? - Suppose the polarizing axis of the analyzer makes an angle θ with the polarizing axis of the first polarizer. - We can resolve the linearly polarized light that is transmitted by the first polarizer into two components, as shown in Fig. 33.24, one parallel and the other perpendicular to the axis of the analyzer. - Only the parallel component, with amplitude Ecosθ, is transmitted by the analyzer. - The transmitted intensity is greatest when θ = 0, and it is zero when the polarizer and analyzer are crossed.. ## **Polarization by Reflection** - Unpolarized light can be polarized, either partially or totally, by reflection. - In Fig. 33.26, unpolarized natural light is incident on a reflecting surface between two transparent optical materials. - For most angles of incidence, waves for which the electric-field vector *E* is perpendicular to the plane of incidence (that is, parallel to the reflecting surface) are reflected more strongly than those for which *E* lies in this plane. - In this case the reflected light is partially polarized in the direction perpendicular to the plane of incidence. - But at one particular angle of incidence, called the polarizing angle θ*p*, the light for which *E* lies in the plane of incidence is not reflected at all but is completely refracted. - At this same angle of incidence the light for which *E* is perpendicular to the plane of incidence is partially reflected and partially refracted. - The reflected light is therefore completely polarized perpendicular to the plane of incidence, as shown in Fig. 33.26. - The refracted (transmitted) light is partially polarized parallel to this plane; the refracted light is a mixture of the component parallel to the plane of incidence, all of which is refracted, and the remainder of the perpendicular component. - In 1812 the British scientist Sir David Brewster discovered that when the angle of incidence is equal to the polarizing angle θ*p*, the reflected ray and the refracted ray are perpendicular to each other. - In this case the angle of refraction θ*r* equals 90 - θ*p*. - From the law of refraction, $n_a sin \theta_p = n_b sin \theta_r = n_b sin(90^o - \theta_p) = n_b cos\theta_p$ Since (sin θ*p*)/(cos θ*p*) = tan θ*p*, we can rewrite this equation as $\tan \theta_p = \frac{n_b}{n_a}$ - This relationship is known as Brewster's law. - Although discovered experimentally, it can also be derived from a wave model by using Maxwell's equations. - Polarization by reflection is the reason polarizing filters are widely used in sunglasses. - When sunlight is reflected from a horizontal surface, the plane of incidence is vertical, and the reflected light contains a preponderance of light that is polarized in the horizontal direction. - When the reflection occurs at a smooth asphalt road surface, it causes unwanted glare. - To eliminate this glare, the polarizing axis of the lens material is made vertical, so very little of the horizontally polarized light reflected from the road is transmitted to the eyes. - The glasses also reduce the overall intensity of the transmitted light to somewhat less than 50% of the intensity of the unpolarized incident light. ## 33.6 **Scattering of Light** - The sky is blue. - Sunsets are red. - Skylight is partially polarized; that's why the sky looks darker from some angles than from others when it is viewed through Polaroid sunglasses. - As we will see, a single phenomenon is responsible for all of these effects. - When you look at the daytime sky, the light that you see is sunlight that has been absorbed and then re-radiated in a variety of directions. - This process is called scattering. - If the earth had no atmosphere, the sky would appear as black in the daytime as it does at night, just as it does to an astronaut in space or on the moon. - Figure 33.31 shows some of the details of the scattering process. - Sunlight, which is unpolarized, comes from the left along the x-axis and passes over an observer looking vertically upward along the y-axis. - We are viewing the situation from the side. - Consider the molecules of the earth's atmosphere located at point O. - The electric field in the beam of sunlight sets the electric charges in these molecules into vibration. - Since light is a transverse wave, the direction of the electric field in any component of the sunlight lies in the yz-plane, and the motion of the charges takes place in this plane. - There is no field, and hence no motion of charges, in the direction of the x-axis. - An incident light wave sets the electric charges in the molecules at point O vibrating along the line of E. - We can resolve this vibration into two components, one along the y-axis and the other along the z-axis. - Each component in the incident light produces the equivalent of two molecular "antennas," oscillating with the same frequency as the incident light and lying along the y- and z-axes. - We mentioned in Chapter 32 that an oscillating charge, like those in an antenna, does not radiate in the direction of its oscillation. - Thus the "antenna" along the y-axis does not send any light to the observer directly below it, although it does emit light in other directions. - Therefore the only light reaching this observer comes from the other molecular "antenna," corresponding to the oscillation of charge along the z-axis. - This light is linearly polarized, with its electric field along the z-axis (parallel to the "antenna"). - The red vectors on the y-axis below point O in Fig. 33.31 show the direction of polarization of the light reaching the observer. - As the original beam of sunlight passes though the atmosphere, its intensity decreases as its energy goes into the scattered light. - Detailed analysis of the scattering process shows that the intensity of the light scattered from air molecules increases in proportion to the fourth power of the frequency (inversely to the fourth power of the wavelength). - Thus the intensity ratio for the two ends of the visible spectrum is (750 nm/380 nm) = 15. - Roughly speaking, scattered light contains 15 times as much blue light as red, and that's why the sky is blue. - When the sunbathing observer on the left looks up, he sees blue, polarized sunlight that has been scattered by air molecules. - The observer on the right sees reddened, unpolarized light when he looks at the sun. ## 33.7 **Huygens's Principle** - The laws of reflection and refraction of light rays, as introduced in Section 33.2, were discovered experimentally long before the wave nature of light was firmly established. - However, we can derive these laws from wave considerations and show that they are consistent with the wave nature of light. - We begin with a principle called Huygens's principle. - This principle, stated originally by the Dutch scientist Christiaan Huygens in 1678, is a geometrical method for finding, from the known shape of a wave front at some instant, the shape of the wave front at some later time. - Huygens assumed that every point of a wave front may be considered the source of secondary wavelets that spread out in all directions with a speed equal to the speed of propagation of the wave. - The new wave front at a later time is then found by constructing a surface tangent to the secondary wavelets or, as it is called, the envelope of the wavelets. - All the results that we obtain from Huygens's principle can also be obtained from Maxwell's equations, but Huygens's simple model is easier to use. - Figure 33.33 illustrates Huygens's principle. - The original wave front AA' is traveling outward from a source, as indicated by the arrows. - We want to find the shape of the wave front after a time interval t. - We assume that v, the speed of propagation of the wave, is the same at all points. - Then in time t the wave front travels a distance vt. - We construct several circles (traces of spherical wavelets) with radius r = vt, centered at points along AA'. - The trace of the envelope of these wavelets, which is the new wave front, is the curve BB' ## **Reflection and Huygens's Principle** - To derive the law of reflection from Huygens's principle, we consider a plane wave approaching a plane reflecting surface. - In Fig. 33.34a the lines AA', OB', and NC' represent successive positions of a wave front approaching the surface MM'. - Point A on the wave front AA' has just arrived at the reflecting surface. - We can use Huygens's principle to find the position of the wave front after a time interval t. - With points on AA' as centers, we draw several secondary wavelets with radius vt. - The wavelets that originate near the upper end of AA' spread out unhindered, and their envelope gives the portion OB' of the new wave front. - If the reflecting surface were not there, the wavelets originating near the lower end of AA' would similarly reach the positions shown by the broken circular arcs. - Instead, these wavelets strike the reflecting surface. - The effect of the reflecting surface is to change the direction of travel of those wavelets that strike it, so the part of a wavelet that would have penetrated the surface actually lies to the left of it, as shown by the full lines. - The first such wavelet is centered at point A; the envelope of all such reflected wavelets is the portion OB of the wave front. - The trace of the entire wave front at this instant is the bent line BOB'. - A similar construction gives the line CNC' for the wave front after another interval t. - From plane geometry the angle θi between the incident wave front and the surface is the same as that between the incident ray and the normal to the surface and is therefore the angle of incidence. - Similarly, θr is the angle of reflection. - To find the relationship between these angles, we consider Fig. 33.34b. - From O we draw OP = vt, perpendicular to AA'. - Now OB, by construction, is tangent to a circle of radius vt with center at A. - If we draw AQ from A to the point of tangency, the triangles APO and OQA are congruent because they are right triangles with the side AO in common and with AQ = OP = vt. - The angle θr therefore equals the angle θi, and we have the law of reflection. ## **Refraction and Huygens's Principle** - We can derive the law of refraction by a similar procedure. - In Fig. 33.35a we consider a wave front, represented by line AA', for which point A has just arrived at the boundary surface SS' between two transparent materials a and b, with indexes of refraction and wave speeds v*a* and v*b*. - (The reflected waves are not shown; they proceed as in Fig. 33.34.) - We can apply Huygens's principle to find the position of the refracted wave fronts after a time t. - With points on AA' as centers, we draw several secondary wavelets. - Those originating near the upper end of AA' travel with speed a and, after a time interval t, are spherical surfaces of radius v*a*t. - The wavelet originating at point A, however, is traveling in the second material b with speed v*b* and at time t is a spherical surface of radius v*b*t. - The envelope of the wavelets from the original wave front is the plane whose trace