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 electro-magnetic waves in vacuum are the same, and is equal to: 882.99792458 10 m/s 3 10 m/sc =×≈ ×. 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 corres-ponding frequency range is from about 430 THz (red limit) to about 750 THz (violet limit). (THz stands for terahertz, or 1012 Hertz.) When the length scales of all objects, gaps, and holes involved are much lar-ger 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 reflected beam surface normal incident beam Plane of incidence reflecting surface θ r θ i 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 -The speed of light in an optical material, 1/εµ=v, is slower than that in vacuum, 001/cεµ=, by the factor mnKK= ,where 0/Kεε≡ is called the dielectric constant, and 0/mKµµ≡ 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 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: 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). refracted beam surface normal reflected beam flat interface θ a θ b n b n a incident beam - 2 -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/nb 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 differ-ent 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 violetred water dropletredviolet sun light - 3 -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: EGEG or These are the two possible linear (or planar) polarizations of an electro-magnetic (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 vec-tor is decomposed into x- and y- components: This is because an electric field is a vector. EG y x θ A Polaroid sheet has an intrinsic direction, It allows only light polarized in this direction to get through. Light from ordinary sources like a fluorescent or incandescent light bulb is unpolarized, meaning that it is an incoherent mixture of both polarizations of equal intensities. A Polaroid sheet is a polarizer, in the sense that un-polarized light, after going through a Polaroid sheet, becomes