Chapter 33 The Nature and Propagation of Light by C.-R. Hu
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1 Chapter 33 The Nature and Propagation of Light by C.-R. Hu Light is a transverse wave of the electromagnetic field. In 1873, James C. Maxwell predicted it from the Maxwell equations. The speed of all electromagnetic waves in vacuum are the same, and is equal to: 8 8 c = m/s 3 10 m/s. Electromagnetic waves become visible to a human eye if its wavelength lies from about 400 nm (violet limit) to about 700 nm (red limit). The corresponding frequency range is from about 430 THz (red limit) to about 750 THz (violet limit). (THz stands for terahertz, or Hertz.) When the length scales of all objects, gaps, and holes involved are much larger than the light wavelength, one can treat light as made of light rays perpendicular to the wave fronts (where the crests and troughs of the wave are located). Light rays travel in straight lines in a uniform medium. They change direction only at the interface of two different optical media of different indices of refraction (n), giving rise to reflection and refraction. Light rays will be curved if n changes continuously. It can be viewed as many mini-refractions, with little reflection in each step. 1. The law of reflection incident beam surface normal reflected beam θ i θ r Plane of incidence reflecting surface a. The plane of incidence is a plane containing the incident beam and the surface normal. b. The reflected beam lies in the plane of incidence and is on the other side of the surface normal as the incident beam. c. θ i = θ r where θ i is the angle of incidence, or the incident angle; θ r is the angle of reflection. 2. Index of refraction - 1 -
2 The speed of light in an optical material, v = 1/ εµ, is slower than that in vacuum, c= 1/ ε 0µ 0, by the factor n= KKm,where K ε / ε 0 is called the dielectric constant, and K m µ / µ 0 is called the relative permeability. n is called the index of refraction of an optical material, so that v = c / n (speed of light in an optical material). Most materials are non-magnetic, with K m = 1. Also, most materials have K > 1. Thus most materials have n > 1, and v < c. 3. The law of refraction, or Snell s law incident beam surface normal reflected beam n a θ a flat interface n b refracted beam When a light beam, initially propagating in a medium with an index of refraction n 1, is incident on a flat interface with a different medium of an index of refraction n 2, then the light beam will be partially reflected in accordance with the law of reflection, and partially transmitted into the second medium with a change of direction. This is called refraction. The law of refraction, or Snell s law, is made of two statements: θ b Statement 1: The incident beam, the reflected beam, the refracted beam, and the surface normal, all lie in one plane, which is the plane of incidence. Statement 2: Let θ a be the angle of incidence (i.e., the angle between the incident beam and the surface normal), and let θ b be the angle of refraction (i.e., the angle between the refracted beam and the surface normal), then these two angles are related by the equation: n a sinθ a = n b sinθ b (the law of refraction, or Snell s law)
3 4. Consequences of the Snell s law (4a) When light crosses an interface between two different optical media, its frequency does not change, but its wavelength changes, since λ a = v a/f = c/na f, and λ b = v b /f = c/n b f. Clearly, λ a = λ / n a, and λ b = λ / n b, where λ = c / f is the wavelength of light in vacuum with the same frequency. (4b) If n a > n b, that is, if light is incident from a denser medium, then we can satisfy sin θ a = n b / n a, which implies sin θ b = 1, or θ b = 90. This particular θ a is called the critical angle for total reflection. That is: sin θ crit = n b / n a (the critical angle for total reflection). Clearly, this critical angle exists only on the side with the larger n, so that n a > n b, because sin θ crit is always less than unity. For θ a > θ crit, refraction is no longer possible, so the incident beam becomes 100% reflected. This is what is meant by total reflection. Diamond has the largest n (= 2.417). So it can be cut easily into a shape to give a great deal of total reflection on its back side. This explains diamond jewelry s brilliance. Other applications of total reflection include totally-reflecting prisms inside binoculars, and optic fibers. (4c) The index of refraction n varies with frequency of light slightly. This phenomenon is called dispersion. This fact can be used to separate different frequency components of light using a prism. It also explains the origin of rainbows (water droplets separating the different frequency components of sunlight). Mixture of frequencies n(f ) prism red violet sun light violet red water droplet - 3 -
4 Sun light and lamp light are examples of white light, which refers to light which has practically all visible and lower frequency components, and usually also a range of higher frequency components. Optical fiber uses total internal reflection to make light follow the path of a curved fiber. 5. Polarization Light is a transverse wave. That is, the electric and magnetic fields in a light wave are both perpendicular to the direction of the light wave. It can then have two mutually perpendicular choices: E E or These are the two possible linear (or planar) polarizations of an electromagnetic (or light) wave. (The first direction is arbitrarily chosen, as long as it is perpendicular to the propagation direction of the wave. The second direction must be perpendicular to the first and to the propagation direction of the wave.) A light wave linearly-polarized in any other direction can be decomposed into these two linear polarizations, in the same wave that a vector is decomposed into x- and y- components: This is because an electric field is a vector. A Polaroid sheet has an intrinsic direction, It allows only light polarized in this direction to get through. E Light from ordinary sources like a fluorescent or incandescent light bulb is unpolarized, meaning θ x that it is an incoherent mixture of both polarizations of equal intensities. A Polaroid sheet is a polarizer, in the sense that unpolarized light, after going through a Polaroid sheet, becomes polarized in the direction determined by the intrinsic direction of the Polaroid sheet, and its intensity is cut down by a factor of 2 (because the Polaroid sheet has blocked the other polarization). (The blocking is not perfect in most commercial Polaroid sheets.) They are called polarization filters. If the incoming light is already linearly polarized in the direction Ê 1 (a unit vector in the direction of E 1 ), and it is sent through a Polaroid sheet with an intrinsic direction (another unit vector), Then the electric field of the Ê 2 y - 4 -
5 incoming light in the direction must be projected into the direction Ê, Ê1 2 leading to a factor Ê1 Ê2 = cos θ, where θ is the angle between the two unit vectors. As light intensity is proportional to the square of the electric field, the light coming out of the Polaroid sheet will be cut down by a factor cos from the intensity of the incoming light. Hence the formula I = I cos 2 θ. 0 If the out-coming light is sent through another Polaroid sheet with an intrinsic direction Ê 3, Then the electric field will be cut down by another factor Ê 2 Ê 3 = cos θ ', because the electric field is projected again. The intensity is then cut down by another factor of cos 2 θ '. The direction of polarization of the out-coming light is always determined by the intrinsic direction of the last Polaroid sheet it has gone through. Light reflected from a surface will in general be partially polarized in the direction perpendicular to the incident plane. (For incident angle θ i = 0, the reflected light is unpolarized. For the special incident angle θ p obeying tan θ p = n 2 / n 1, (Malus s law) 2 θ (the Brewster s law) the reflected light is 100% polarized in the direction perpendicular to the incident plane. This special incident angle is called the Brewster angle, and has the notation θ p, since it is also called the polarization angle. It can be easily shown that, at this angle, the refracted light is perpendicular to the reflected light. (Simply use the above equation and the Snell's law.) Applications: 1. Polarized sun glasses to eliminate reflected glare (because reflected light is mostly polarized with electric field perpendicular to the incident plane); 2. liquid crystal display (LCD), which is used as, for example, computer screen. So far the electric field vector oscillates in a fixed direction perpendicular to the light direction. That is called linear or planar polarization. But two electromagnetic waves can occupy the same location without interferring each other. (This is called the principle of superposition.) So one can consider two mutually-perpendicular linearly polarized light waves occupying the same location. The electric fields of these two waves should then be added up vectorially (i.e., as vectors). Assuming that the wavelengths and - 5 -
6 directions of propagation of these two waves are the same, the result of the vector sum will depend on their amplitudes and the relative phase of the two waves. If the two waves have the same amplitude and are in phase, the result is a wave linearly polarized in the direction which is at 45 with the original two. E 1 E 2 But if the two waves are 90 out of phase, and still have the same amplitudes, then they add up to a circularly-polarized light wave: E 1 E 2 That is, it you look against the direction of the wave, the electric field vector either rotates clockwise, called right circularly polarized wave, or counterclockwise, called left circularly polarized wave. If the amplitudes of the two waves are not equal, or if the relative phase of the two waves is neither zero nor 90, then the tip of the electric field vector will draw an ellipse rather than a circle, and is called an elliptically polarized wave. Of course, the ellipse can also appear as: or, etc., - 6 -
7 and the electric field vector can again rotate in either directions (with its length changing at the same time). In all cases discussed above, only the electric field vector has been drawn. The magnetic field vector is always present, and is always related to the direction of propagation and the electric field vector by the right hand rule: E B the direction of propagation Unpolarized light, after scattering by an atom or a dust particle, also becomes at least partially polarized. If the scattering angle is 90, then the scattered light is purely linearly polarized, in the direction perpendicular to the scattering plane (the plane containing the directions of the incoming light and the scattered light): the scattering plane because the other direction of the electric field in the incoming wave is now along the direction of the scattered wave, and is unacceptable, because the light wave is a transverse wave. 6. Huygen s principle This principle is a general property of all waves, and can be easily demonstrated with water or sound wave. Imaging that a planar sound wave with straight wave fronts is generated on the left of a long straight corridor. The wave propagates to the right but is stopped by a wall with a small hole at its center. Then to the right of the wall you will have a spherical sound wave emanating from the hole in the wall. This is the essence of the Huygen s principle. wave front at a earlier time wave front at a later time - 7 -
8 Namely, any point of a wave front is the source of a spherical wave. If there is no wall present, then indeed every point of a wave front is the source of a spherical wave, and all these spherical waves emanated from all points of the same wave front define a new wave front by their envelop. If the radii of all of these spherical waves are equal to the speed of the wave at the respective locations times t, then the new wave front obtained from the envelop is the wave front at time t later. When a plane EM wave reaches an interface of two different optical media at an angle θ 1, one end of a wave front touches the interface first. Since the speed of light is different in the two media (c/n 1 vs. c/n 2 ), the radii of spherical waves drawn on the two ends of that wave front are therefore different. (In the figure shown, n 2 > n 1, so the spherical wave drawn on the left is smaller, since it is in the lower medium in which the speed of light c/n 2 is smaller.) As a result the direction of light becomes refracted when light crosses the interface between the two media. From the figure it can be seen that sin θ 1 = (c/n 1 ) t /d, n 1 θ 2 n 2 θ 1 θ 1 θ 2 (This figure assumes n > n.) where d is the distance between the two ends of the wave front, and sin θ 2 = (c/n 2 ) t /d. Thus one obtains the Snell s law: n 1 sin θ 1 = n 2 sin θ 2 for refraction. (Both are equal to c t/ d.) 2 1 If n varies in the direction perpendicular to the direction of propagation of the light wave, then the successive wave fronts are not parallel to each other, leading to curved light rays and path: because ( c/ n) t n decreases is longer for smaller n. This is how mirage is formed
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