and the behavior of p-n junctions in solid-state rectifiers. The invention of the laser extended interest in nonlinear effects to the realm of optics, where phenomena such as parametric amplification and oscillation, harmonic generation, and magneto-optic inter actions have found Provided that the external field applied to a nonlinear electric material is small com- pared to the internal molecular fields, the relationship between E and D can be expanded sIsOIansAy ou Bu!!q: xe [euronet idonostuB UB od 'PIay oL4Da1a a+ Jo soles o/4e,aya ts, the constitutive relation is[131] D(r,)=60E(,1)+∑x1E1(,1)+∑xE,DE(r1)+ +∑xE1,1)Er,DE(r,1)+ where the index i= 1, 2, 3 refers to the three components of the fields D and E. The first sum in(2.43)is identical to the constitutive relation for linear anisotropic materi- als.Thus,x) is identical to the susceptibility dyadic of a linear anisotropic medium considered earlier. The quantity Xijk is called the second-order susceptibility, and is a three-dimensional matrix (or third rank tensor)describing the nonlinear electric effects quadratic in E. Similarly xii is called the third-order susceptibility, and is a four- dimensional matrix (or fourth rank tensor) describing the nonlinear electric effects cubic in E. Numerical values of xik and xiikl are given in Shen [174 for a variety of crystals When the material shows hysteresis effects, D at any point r and time t is due not only to the value of e at that point and at that time, but to the values of e at all points and times. That is, the material displays both temporal and spatial dispersion 2. 3 Maxwell,'s equations in moving frames The essence of special relativity is that the mathematical forms of Maxwells equa tions are identical in all inertial reference frames: frames moving with uniform velocities relative to the laboratory frame of reference in which we perform our measurements. This form invariance of Maxwells equations is a specific example of the general physical principle of covariance. In the laboratory frame we write the differential equations of Maxwell's theory db(r, t) V×E(r,t) V×H(r,t)=J(r,n)+ aD(r, t) V·D(r,t)=p(r,) V·B(r,t)=0 r J(r, t) @2001 by CRC Press LLC
and the behavior of p-n junctions in solid-state rectifiers. The invention of the laser extended interest in nonlinear effects to the realm of optics, where phenomena such as parametric amplification and oscillation, harmonic generation, and magneto-optic interactions have found applications in modern devices [174]. Provided that the external field applied to a nonlinear electric material is small compared to the internal molecular fields, the relationship between E and D can be expanded in a Taylor series of the electric field. For an anisotropic material exhibiting no hysteresis effects, the constitutive relation is [131] Di(r, t) = 0Ei(r, t) + 3 j=1 χ(1) i j E j(r, t) + 3 j,k=1 χ(2) ijk E j(r, t)Ek (r, t) + + 3 j,k,l=1 χ(3) ijkl E j(r, t)Ek (r, t)El(r, t) +··· (2.43) where the index i = 1, 2, 3 refers to the three components of the fields D and E. The first sum in (2.43) is identical to the constitutive relation for linear anisotropic materials. Thus, χ(1) i j is identical to the susceptibility dyadic of a linear anisotropic medium considered earlier. The quantity χ(2) ijk is called the second-order susceptibility, and is a three-dimensional matrix (or third rank tensor) describing the nonlinear electric effects quadratic in E. Similarly χ(3) ijkl is called the third-order susceptibility, and is a fourdimensional matrix (or fourth rank tensor) describing the nonlinear electric effects cubic in E. Numerical values of χ(2) ijk and χ(3) ijkl are given in Shen [174] for a variety of crystals. When the material shows hysteresis effects, D at any point r and time t is due not only to the value of E at that point and at that time, but to the values of E at all points and times. That is, the material displays both temporal and spatial dispersion. 2.3 Maxwell’s equations in moving frames The essence of special relativity is that the mathematical forms of Maxwell’s equations are identical in all inertial reference frames: frames moving with uniform velocities relative to the laboratory frame of reference in which we perform our measurements. This form invariance of Maxwell’s equations is a specific example of the general physical principle of covariance. In the laboratory frame we write the differential equations of Maxwell’s theory as ∇ × E(r, t) = −∂B(r, t) ∂t , ∇ × H(r, t) = J(r, t) + ∂D(r, t) ∂t , ∇ · D(r, t) = ρ(r, t), ∇ · B(r, t) = 0, ∇ · J(r, t) = −∂ρ(r, t) ∂t
Figure 2.1: Primed coordinate system moving with velocity v relative to laborator (unprimed) coordinate system Similarly, in an inertial frame having four-dimensional coordinates(r, t) we have V"×E(r,t)= at' v×H(r,t)=J(r,t)+ aD(r, t at V. D(r, t)=p(r, t), v·B(r,t)=0 V′·J(r,t') dp(r,t The primed fields measured in the moving system do not have the same numerical values as the unprimed fields measured in the laboratory. To convert between E and e, B and B, and so on, we must find a way to convert between the coordinates(r, t) and(r, t') 2.3.1 Field conversions under galilean transformation We shall assume that the primed coordinate system moves with constant velocity v relative to the laboratory frame(Figure 2.1). Prior to the early part of the twentieth century, converting between the primed and unprimed coordinate variables was intuitive and obvious: it was thought that time must be measured identically in each coordinate system, and that the relationship between the space variables can be determined simply by the displacement of the moving system at time t= t'. Under these assumptions, and under the further assumption that the two systems coincide at time t=0, we can write t=t. r This is called a Galilean transformation. We can use the chain rule to describe the manner in which differential operations transform, i.e., to relate derivatives with respect to the laboratory coordinates to derivatives with respect to the inertial coordinates. We aar′ a ax' a dy' d.az′a at ar' at d dt dy at dz @2001 by CRC Press LLC
Figure 2.1: Primed coordinate system moving with velocity v relative to laboratory (unprimed) coordinate system. Similarly, in an inertial frame having four-dimensional coordinates (r , t ) we have ∇ × E (r , t ) = −∂B (r , t ) ∂t , ∇ × H (r , t ) = J (r , t ) + ∂D (r , t ) ∂t , ∇ · D (r , t ) = ρ (r , t ), ∇ · B (r , t ) = 0, ∇ · J (r , t ) = −∂ρ (r , t ) ∂t . The primed fields measured in the moving system do not have the same numerical values as the unprimed fields measured in the laboratory. To convert between E and E , B and B , and so on, we must find a way to convert between the coordinates (r, t) and (r , t ). 2.3.1 Field conversions under Galilean transformation We shall assume that the primed coordinate system moves with constant velocity v relative to the laboratory frame (Figure 2.1). Prior to the early part of the twentieth century, converting between the primed and unprimed coordinate variables was intuitive and obvious: it was thought that time must be measured identically in each coordinate system, and that the relationship between the space variables can be determined simply by the displacement of the moving system at time t = t . Under these assumptions, and under the further assumption that the two systems coincide at time t = 0, we can write t = t, x = x − vx t, y = y − vy t, z = z − vzt, or simply t = t, r = r − vt. This is called a Galilean transformation. We can use the chain rule to describe the manner in which differential operations transform, i.e., to relate derivatives with respect to the laboratory coordinates to derivatives with respect to the inertial coordinates. We have, for instance, ∂ ∂t = ∂t ∂t ∂ ∂t + ∂x ∂t ∂ ∂x + ∂y ∂t ∂ ∂y + ∂z ∂t ∂ ∂z
a (VV) Similarly from which V×A(r,t)=V"×A(r,t),V.A(r,t)=V′A(r,t) for each vector field A Newton was aware that the laws of mechanics are invariant with respect to galilean transformations. Do Maxwell's equations also behave in this way? Let us use the galilean transformation to determine which relationship between the primed and unprimed fields results in form invariance of Maxwell, s equations. We first examine VxE, the spatial rate of change of the laboratory field with respect to the inertial frame spatial coordinates VxE=V×E=-=、0 aB +(v·V)B by(2.45)and(2.44). Rewriting the last term by(B45) we hav (v·V)B=-V×(v×B) since y is constant and v.b=v.b=o. hence V"×(E+V×B)= (2.46) Similarly aD V"×H=V×H=J+一=J++V×(v×D)-v(VD) where V. D=V.D=p so that J=v.J=-20=-0 +(v·V)p and we may use(B42)to write (v·V)p=v·(Vp)=V·(pv) btaining v)= @2001 by CRC Press LLC
= ∂ ∂t − vx ∂ ∂x − vy ∂ ∂y − vz ∂ ∂z = ∂ ∂t − (v · ∇ ). (2.44) Similarly ∂ ∂x = ∂ ∂x , ∂ ∂y = ∂ ∂y , ∂ ∂z = ∂ ∂z , from which ∇ × A(r, t) = ∇ × A(r, t), ∇ · A(r, t) = ∇ · A(r, t), (2.45) for each vector field A. Newton was aware that the laws of mechanics are invariant with respect to Galilean transformations. Do Maxwell’s equations also behave in this way? Let us use the Galilean transformation to determine which relationship between the primed and unprimed fields results in form invariance of Maxwell’s equations. We first examine ∇ ×E, the spatial rate of change of the laboratory field with respect to the inertial frame spatial coordinates: ∇ × E =∇× E = −∂B ∂t = −∂B ∂t + (v · ∇ )B by (2.45) and (2.44). Rewriting the last term by (B.45) we have (v · ∇ )B = −∇ × (v × B) since v is constant and ∇ · B =∇· B = 0, hence ∇ × (E + v × B) = −∂B ∂t . (2.46) Similarly ∇ × H =∇× H = J + ∂D ∂t = J + ∂D ∂t + ∇ × (v × D) − v(∇ · D) where ∇ · D =∇· D = ρ so that ∇ × (H − v × D) = ∂D ∂t − ρv + J. (2.47) Also ∇ · J =∇· J = −∂ρ ∂t = − ∂ρ ∂t + (v · ∇ )ρ and we may use (B.42) to write (v · ∇ )ρ = v · (∇ ρ) = ∇ · (ρv), obtaining ∇ · (J − ρv) = − ∂ρ ∂t . (2.48)
Equations(2.46),(2.47), and(2.48)show that the forms of Maxwell's equations in the ertial and laboratory frames are identical provided that E'=E+v×B, H=H-×D 2222 J=J-pv 01235 That is,(2.49 )-(2.54)result in form invariance of Faraday's law, Ampere's law, an continuity equation under a Galilean transformation. These equations express the measured by a moving observer in terms of those measured in the laboratory frame. To convert the opposite way, we need only use the principle of relativity. Neither observer can tell whether he or she is stationary -only that the other observer is moving relative to him or her. To obtain the fields in the laboratory frame we simply change the sign on v and swap primed with unprimed fields in(2.49)-(2. 54) E=E-v×B' H=H+v×D, B (258) J=J+pv (260) According to(2.53), a moving observer interprets charge stationary in the laboratory frame as an additional current moving opposite the direction of his or her motion. This seems reasonable. However, while e depends on both E and B, the field B is unchange under the transformation. Why should B have this special status? In fact, we may uncover an inconsistency among the transformations by considering free space where 2.22)and(2.23) hold: in this case(2.49) gives D/0=D/∈o+v×poH rather than(2.50). Similarly, from(2.51)we get B=B instead of (2.52). Using these, the set of transformations becomes B D=D+v×H/c2, E/ 2366 @2001 by CRC Press LLC
Equations (2.46), (2.47), and (2.48) show that the forms of Maxwell’s equations in the inertial and laboratory frames are identical provided that E = E + v × B, (2.49) D = D, (2.50) H = H − v × D, (2.51) B = B, (2.52) J = J − ρv, (2.53) ρ = ρ. (2.54) That is, (2.49)–(2.54) result in form invariance of Faraday’s law, Ampere’s law, and the continuity equation under a Galilean transformation. These equations express the fields measured by a moving observer in terms of those measured in the laboratory frame. To convert the opposite way, we need only use the principle of relativity. Neither observer can tell whether he or she is stationary — only that the other observer is moving relative to him or her. To obtain the fields in the laboratory frame we simply change the sign on v and swap primed with unprimed fields in (2.49)–(2.54): E = E − v × B , (2.55) D = D , (2.56) H = H + v × D , (2.57) B = B , (2.58) J = J + ρ v, (2.59) ρ = ρ . (2.60) According to (2.53), a moving observer interprets charge stationary in the laboratory frame as an additional current moving opposite the direction of his or her motion. This seems reasonable. However, while E depends on both E and B , the field B is unchanged under the transformation. Why should B have this special status? In fact, we may uncover an inconsistency among the transformations by considering free space where (2.22) and (2.23) hold: in this case (2.49) gives D / 0 = D/ 0 + v × µ0H or D = D + v × H/c2 rather than (2.50). Similarly, from (2.51) we get B = B − v × E/c2 instead of (2.52). Using these, the set of transformations becomes E = E + v × B, (2.61) D = D + v × H/c2 , (2.62) H = H − v × D, (2.63) B = B − v × E/c2 , (2.64) J = J − ρv, (2.65) ρ = ρ. (2.66)
These can also be written using dyadic notation as E′=1.E+B·(CB), (267) β·E+I·(cB) H=-·(cD)+IH, 0 -Pz By =|B20 with B=v/c. This set of equations is self-consistent among Maxwells equations. How ever, the equations are not consistent with the assumption of a Galilean transformation of the coordinates, and thus Maxwells equations are not covariant under a galilean transformation. Maxwells equations are only covariant under a Lorentz transforma tion as described in the next section. Expressions(2.61)-(2.64) turn out to be accurate to order u/c, hence are the results of a first-order lorentz transformation. Only wher v is an appreciable fraction of c do the field conversions resulting from the first-order Lorentz transformation differ markedly from those resulting from a Galilean transforma tion; those resulting from the true Lorentz transformation require even higher velocities to differ markedly from the first-order expressions. Engineering accuracy is often accom- plished using the Galilean transformation. This pragmatic observation leads to quite a bit of confusion when considering the large-scale forms of Maxwell's equations, as we shall soon see 2.3.2 Field conversions under lorentz transformation To find the proper transformation under which Maxwell's equations are covariant ye must discard our notion that time progresses the same in the primed and the un- primed frames. The proper transformation of coordinates that guarantees covariance of Maxwells equations is the Lorentz transformation r=yct-y·r, (271) Bct, he 1-p2 a=1+(y-1),,B=1( This is obviously more complicated than the Galilean transformation; only as B-0 are the Lorentz and Galilean transformations equivalent Not surprisingly, field conversions between inertial reference frames are more com- plicated with the Lorentz transformation than with the Galilean transformation. For simplicity we assume that the velocity of the moving frame has only an x-component v= &u. Later we can generalize this to any direction. Equations(2.71)and(2.72) x'=x+(y-l)x-yut @2001 by CRC Press LLC
These can also be written using dyadic notation as E = ¯ I · E + β¯ · (cB), (2.67) cB = −β¯ · E + ¯ I · (cB), (2.68) and cD = ¯ I · (cD) + β¯ · H, (2.69) H = −β¯ · (cD) + ¯ I · H, (2.70) where [β¯ ] = 0 −βz βy βz 0 −βx −βy βx 0 with β = v/c. This set of equations is self-consistent among Maxwell’s equations. However, the equations are not consistent with the assumption of a Galilean transformation of the coordinates, and thus Maxwell’s equations are not covariant under a Galilean transformation. Maxwell’s equations are only covariant under a Lorentz transformation as described in the next section. Expressions (2.61)–(2.64) turn out to be accurate to order v/c, hence are the results of a first-order Lorentz transformation. Only when v is an appreciable fraction of c do the field conversions resulting from the first-order Lorentz transformation differ markedly from those resulting from a Galilean transformation; those resulting from the true Lorentz transformation require even higher velocities to differ markedly from the first-order expressions. Engineering accuracy is often accomplished using the Galilean transformation. This pragmatic observation leads to quite a bit of confusion when considering the large-scale forms of Maxwell’s equations, as we shall soon see. 2.3.2 Field conversions under Lorentz transformation To find the proper transformation under which Maxwell’s equations are covariant, we must discard our notion that time progresses the same in the primed and the unprimed frames. The proper transformation of coordinates that guarantees covariance of Maxwell’s equations is the Lorentz transformation ct = γ ct − γβ · r, (2.71) r = α¯ · r − γβct, (2.72) where γ = 1 1 − β2 , α¯ = ¯ I + (γ − 1) ββ β2 , β = |β|. This is obviously more complicated than the Galilean transformation; only as β → 0 are the Lorentz and Galilean transformations equivalent. Not surprisingly, field conversions between inertial reference frames are more complicated with the Lorentz transformation than with the Galilean transformation. For simplicity we assume that the velocity of the moving frame has only an x-component: v = xˆv. Later we can generalize this to any direction. Equations (2.71) and (2.72) become x = x + (γ − 1)x − γvt, (2.73)