Chapter 7. Plane Electromagnetic Waves and Wave Propagation 7.1 Plane Monochromatic Waves in Nonconducting Media One of the most important consequences of the Maxwell equations is the equations for electromagnetic wave propagation in a linear medium. In the absence of free charge and current densities the Maxwell equations are (7.1) The wave equations for and are derived by taking the curl of and (7.2) For uniform isotropic linear media we have and , where and are in general complex functions of frequency . Then we obtain (7.3) Since ( ) and, similarly, , (7.4) Monochromatic waves may be described as waves that are characterized by a single frequency. Assuming the fields with harmonic time dependence , so that ( ) ( ) and ( ) ( ) we get the Helmholtz wave equations (7.5) 1 Plane waves in vacuum Suppose that the medium is vacuum, so that and . Further, suppose ( ) varies in only one dimension, say the -direction, and is independent of and . Then Eq. 7.5 becomes ( ) (7.6) ( ) where the wave number . This equation is mathematically the same as the harmonic oscillator equation and has solutions (7.7) ( ) where is a constant vector. Therefore, the full solution is ( ) (7.8) ( ) This represents a sinusoidal wave traveling to the right or left in the -direciton with the speed of light . Using the Fourier superposition theorem, we can construct a general solution of the form ( ) ( ) ( ) (7.9) Plane waves in a nonconducting, nonmagnetic dielectric In a nonmagnetic dielectric, we have and the index of refraction ( ) (7.10) ( ) √ We see that the results are the same as in vacuum, except that the velocity of wave propagation or the phase velocity is now instead of . Then the wave number is (7.11) ( ) ( ) Electromagnetic plane wave of frequency and wave vector Suppose an electromagnetic plane wave with direction of propagation to be constructed, where is a unit vector. Then the variable in the exponent must be replaced by , the projection of in the direction. Thus an electromagnetic plane wave with direction of propagation is described by ( ) (7.12) ( ) where and are complex constant vector amplitudes of the plane wave. and satisfy the wave equations (Eq. 7.5), therefore the dispersion relation is given as ( ) (7.13) 2 Let us substitute the plane wave solutions (Eq. 7.12) into the Maxwell equations. This substitution will impose conditions on the constants, , and , for the plane wave functions to be solutions of the Maxwell equations. For the plane waves, one sees that the operators Thus the Maxwell equations become (7.14) where . The direction and frequency are completely arbitrary. The divergence equations demand that (7.15) This means that and are both perpendicular to the direction of propagation . The magnitude of is determined by the refractive index of the material (7.16) Then is completely determined in magnitude and direction (7.17) √ Note that in vacuum ( ), in SI units. The phase velocity of the wave is . Energy density and flux The time averaged energy density (see Eq. 6.94) is ( ) ( ) This gives | | | | (7.18) The time averaged energy flux is given by the real part of the complex Poynting vector ( ) Thus the energy flow is (7.19) √ | | | | 3 7.2 Polarization and Stokes Parameters There is more to be said about the complex vector amplitudes and . We introduce a right- handed set of orthogonal unit vectors ( ), as shown in Fig. 7.1, where we take to be the propagation direction of the plane wave. In general, the electric field amplitude can be written as (7.20) where the amplitudes and are arbitrary complex numbers. The two plane waves (7.21) with (7.22) (if the index of refraction is real, and have the same phase) are said to be linearly polarized with polarization vectors and . Thus the most general homogeneous plane wave propagating in the direction is expressed as the superposition of two independent plane waves of linear polarization: (7.23) ( ) ( ) Fig 7.1 It is convenient to express the complex components in polar form. Let (7.24) Then, for example, ( ) (7.25) that is, is the phase of the -field component in the -direction. It is no restriction to let (7.26) 4 since merely dictates a certain choice of the origin of . With this choice, ( ) ( ) (7.27) ( ) or the real part is ( ) ( ) ( ) (7.28) The -field is resolved into components in two directions, with real amplitudes and , which may have any values. In addition the two components may be oscillating out of phase by , that is, at any given point , the maximum of in the -direction may be attained at a different time from the maximum of in the -direction. Polarization A detailed picture of the oscillating -field at a certain point, e.g., , is best seen by considering some special cases. ( ) ( ) (7.29) ( ) ( ) Linearly polarized wave If and have the same phase, i.e., , (7.30) ( ) ( ) ( ) ( ) represents a linearly polarized wave, with its polarization vector with ( ) and a magnitude √ , as shown in Fig. 7.2. Fig 7.2 -field of a linearly polarized wave If or , we also have linear polarization. For , (7.31) ( ) ( ) ( ) ( ) is again linearly polarized. 5 Elliptically polarized wave If and have different phases, the wave of Eq. 7.27 is elliptically polarized. The simplest case is circular polarization. Then and : (7.32) ( ) ( ) ( ) At a fixed point in space, the fields are such that the electric vector is constant in magnitude, but sweeps around in a circle at a frequency , as shown in Fig. 7.3. For , ( ), the tip of the -vector traces the circular path counterclockwise. This wave is √ called left circularly polarized (positive helicity) in optics. For , ( ), √ same path but traced clockwise, then the wave is called right circularly polarized (negative helicity). For other values of , we have elliptical polarization for the trace being an ellipse. Fig 7.3 Trace of the tip of the -vector ( ) at a given point in space as a function of time. The propagation direction is point toward us. The traces for and are linearly polarized. The traces for and are left and right circularly polarized, respectively. Stokes Parameters The two circularly polarized waves form a basis set for a general state of polarization. We introduce the complex orthogonal unit vectors: (7.33) ( ) √ 6 They satisfy the orthonormal conditions, (7.34) { Then the most general homogeneous plane wave propagating in the direction (Eq. 7.23) can be expressed as the superposition of two circularly polarized waves: (7.35) ( ) ( ) ( ) where and are complex amplitudes. Fig 7.4 Electric field for an elliptically polarized wave. When the ratio of the amplitudes is expressed as (7.36) the trace of the tip of the -vector is an ellipse as shown in Fig. 7.4. For , the ratio of semimajor to semiminor axis is |( ) ( )|. Stokes parameters The polarization state of the general plane wave (Eq. 7.35) (7.37) ( ) ( ) ( ) can be expressed by either ( ) or ( ). We can determine these complex coefficients using Stokes parameters obtained by intensity measurements using polarizers and wave plates. We express the complex components in polar form: (7.38) The Stokes parameters of the linear polarization basis | | | | (7.39) | | | | [ ] ( ) [ ] ( ) 7 and of the circular polarization basis | | | | (7.40) [ ] ( ) [ ] ( ) | | | | The four parameters are not independent and satisfy the relation (7.41) 7.3 Plane Monochromatic Waves In Conducting Media In a conducting medium there is an induced current density in response to the -field of the wave. The current density J is linearly proportional to the electric field (Ohm’s law, Eq. 5.21): The constant of proportionality is called the conductivity. For an electromagnetic plane wave with direction of propagation (Eq. 7.12) described by ( ) ( ) the Maxwell equation (7.42) becomes ( ) ( ) (7.43) where we define a complex dielectric constant ( ) (7.44) Comparing Eq. 7.44 with Eq. 7.14, we can see that the transverse dispersion relation results in (7.45) √ where we define a complex refractive index (7.46) √ To interpret the wave propagation in the conducting medium, it is useful to express the complex propagation vector as (7.47) 8 Then the plane wave is expressed as ( ) (7.48) ( ) ( ) This is a plane wave propagating in the direction with wavelength ; but it decreases in amplitude, most rapidly in the direction . 7.4 Reflection and Refraction of Electromagnetic Waves at a Plane Interface between Dielectrics Normal Incidence We begin with the simplest possible case: a plane wave normally incident on a plane dielectric interface. We will see that the boundary conditions are satisfied only if reflected and transmitted waves are present. Fig 7.5 Reflection and transmission at normal incidence Fig. 7.5 describes the incident wave ( ) travelling in the z-direction, the reflected wave ( ) travelling in the minus z-direction, and the transmitted wave ( ) travelling in the z- direction. The interface is taken as coincident with the -plane at , with two dielectric media with the indices of refraction, for and for . The electric fields, which are assumed to be linearly polarized in the -direction, are described by ( ) (7.49) ( ) { ( ) where (7.50) From Eq. 7.17, 9 Therefore, the magnetic fields associated with the electric fields of Eq. 7.49 are given by ( ) ( ) (7.51) { ( ) Clearly the reflected and transmitted waves must have the same frequency as the incident wave if boundary conditions at are to be satisfied for all . The -field must be continuous at the boundary, (7.52) The -field must also be continuous, and for nonmagnetic media ( ), so must be the -field: ( ) (7.53) Eqs. 7.52 and 7.53 can be solved simultaneously for the amplitudes and in terms of the incident amplitude : (7.54) The Fresnel coefficients for normal incidence reflection and transmission are defined as (7.55) For , there is a phase reversion for the reflected wave.