BRIEF REVIEW OF ELECTROMAGNETICS Table 1.1 Names and Units of Symbols in Maxwell's Equations Symbol Name Units or Numerical Value E Electric field Vm-1 D Electric flux density Cm-2 Magnetic field Am-1 B Magnetic flux density T (or V-s m-2) 么 Polarization density Cm-2 M Magnetization density A m-1 Current density Am-2 p Charge density Cm-3 % Permittivity of free space 8.85×10-12Fm-1(orCV-1m-1) Permeability of free space 4π×10-7Hm-1(orVs2m-1C-1) charge density p,current density J,polarization density P,and magnetization density M, and in order to include the effect of the fields on the matter,the electric and magnetic flux densities,D and B,are also introduced.Units for these quantities are given in Table 1.1. Maxwell's equations are then written as follows: V.D= (1.1) V.B=0 (1.2) V×E= -aB (1.3) at VxH=J+ aD (1.4) The relations defining D and B are D=80E+P (1.5) B=Lo(H+M) (1.6 The constants so and uo are known as the permittivity and permeability of free space, with the numerical values and units given in Table 1.1.Note also that the symbol o refers to the free charge density (i.e.,any bound charge density associated with the polarization is not included).Similarly,the current density J does not include any currents associated with the motion of bound charges(changes in polarization).In free space we would have p=J=P=M=0. For now we specialize to the case of a linear,isotropic,and source-free medium.By source-free we mean that the charge and current densities are zero (p=0 and J=0).By linear we mean that the medium response (i.e.,the polarization and magnetization)is linear in the applied fields.For the case of the electric field,we write P=80XeE (1.7)
BRIEF REVIEW OF ELECTROMAGNETICS 5 Table 1.1 Names and Units of Symbols in Maxwell’s Equations Symbol Name Units or Numerical Value E Electric field V m−1 D Electric flux density C m−2 H Magnetic field A m−1 B Magnetic flux density T (or V·s m−2 ) P Polarization density C m−2 M Magnetization density A m−1 J Current density A m−2 ρ Charge density C m−3 ε0 Permittivity of free space 8.85 × 10−12 F m−1 (or C V−1 m−1 ) µ0 Permeability of free space 4π × 10−7 H m−1 (or V s2 m−1 C −1 ) charge density ρ, current density J, polarization density P, and magnetization density M, and in order to include the effect of the fields on the matter, the electric and magnetic flux densities, D and B, are also introduced. Units for these quantities are given in Table 1.1. Maxwell’s equations are then written as follows: ∇ · D = ρ (1.1) ∇ · B = 0 (1.2) ∇ × E = −∂B ∂t (1.3) ∇ × H = J + ∂D ∂t (1.4) The relations defining D and B are D = ε0E + P (1.5) B = µ0 (H + M) (1.6) The constants ε0 and µ0 are known as the permittivity and permeability of free space, with the numerical values and units given in Table 1.1. Note also that the symbol ρ refers to the free charge density (i.e., any bound charge density associated with the polarization is not included). Similarly, the current density J does not include any currents associated with the motion of bound charges (changes in polarization). In free space we would have ρ = J = P = M = 0. For now we specialize to the case of a linear, isotropic, and source-free medium. By source-free we mean that the charge and current densities are zero (ρ = 0 and J = 0). By linear we mean that the medium response (i.e., the polarization and magnetization) is linear in the applied fields. For the case of the electric field, we write P = ε0χeE (1.7)
6 INTRODUCTION AND REVIEW where xe is known as the electric susceptibility (dimensionless).Inserting into eq.(1.5), one obtains D 80(1 Xe)E =sE (1.8) The proportionality constant s is termed the dielectric constant,with e=(1+Xe)e0 (1.9) Other common terms include the relative dielectric constant(s/so)and the index of refrac- tion n,which is commonly used in optics,where n2= (1.10) For the case of the magnetic field,we write M=Xm H (1.11) where x is the magnetic polarizability.Using eq.(1.6),we obtain B=o(1+xm)H=uH (1.12) In most cases in ultrafast optics,one is interested in nonmagnetic materials,for which M=0.In this case of zero magnetization,one has B=HoH (1.13) Equations(1.7)and(1.11)are examples of constitutive laws,which specify the response of the material to the fields.The form of these equations as written arises because we have assumed both linear and isotropic media(for nonisotropic media,one would need to replace the assumed scalar susceptibilities with tensors).We note that there are many situations in ultrafast optics where these assumptions are not valid.For example,nonlinear optical effects,which we discuss in later chapters,require by definition that P be a nonlinear function of E. 1.2.2 The Wave Equation and Plane Waves We now consider electromagnetic wave propagation in linear,isotropic,source-free media. To derive the wave equation,we take the curl of eq.(1.3)and insert eq.(1.4),which,using the stated assumptions and a well-known vector identity,gives the following: E Vx VxE=V(V.E)-VE=-ME32 (1.14) I The identity isxxA =V(V.A)-V2A.Note that in Cartesian coordinates V2 has a very simple form,namely V2A =(a-/ax2+a2/ay2+a-/az2)A
6 INTRODUCTION AND REVIEW where χe is known as the electric susceptibility (dimensionless). Inserting into eq. (1.5), one obtains D = ε0(1 + χe)E = εE (1.8) The proportionality constant ε is termed the dielectric constant, with ε = (1 + χe) ε0 (1.9) Other common terms include the relative dielectric constant (ε/ε0) and the index of refraction n, which is commonly used in optics, where n 2 = ε ε0 (1.10) For the case of the magnetic field, we write M = χmH (1.11) where χm is the magnetic polarizability. Using eq. (1.6), we obtain B = µ0 (1 + χm)H = µH (1.12) In most cases in ultrafast optics, one is interested in nonmagnetic materials, for which M = 0. In this case of zero magnetization, one has B = µ0H (1.13) Equations (1.7) and (1.11) are examples of constitutive laws, which specify the response of the material to the fields. The form of these equations as written arises because we have assumed both linear and isotropic media (for nonisotropic media, one would need to replace the assumed scalar susceptibilities with tensors). We note that there are many situations in ultrafast optics where these assumptions are not valid. For example, nonlinear optical effects, which we discuss in later chapters, require by definition that P be a nonlinear function of E. 1.2.2 The Wave Equation and Plane Waves We now consider electromagnetic wave propagation in linear, isotropic, source-free media. To derive the wave equation, we take the curl of eq. (1.3) and insert eq. (1.4), which, using the stated assumptions and a well-known vector identity,1 gives the following: ∇ × ∇ × E = ∇ (∇ · E) − ∇2E = −µε ∂ 2E ∂t2 (1.14) 1 The identity is ∇ × ∇ × A = ∇(∇ · A) − ∇2A. Note that in Cartesian coordinates ∇ 2 has a very simple form, namely ∇ 2A = (∂ 2 /∂x2 + ∂ 2 /∂y2 + ∂ 2 /∂z2 )A
BRIEF REVIEW OF ELECTROMAGNETICS Since V.E =0 also under our conditions,we obtain the wave equation 02E V2E us d12 (1.15) One situation of special interest is the case where the field varies in only one direction, which without loss of generality we take as the z direction.Then the wave equation becomes 子E2E 8k2 (1.16) 8r2 The general solution takes the form E(z,)=1 (t-) (1.17) where Eo is a vector in the x-y plane [eq.(1.1)precludes E from having a z-component]and v=1/Rs.The solution can be verified by plugging back into the wave equation.Equation (1.17)is called a plane-wave solution,since the field does not vary in the transverse (x- y)plane.It also represents a traveling wave,since the field propagates in the z direction without changing its form.In the case of a pulsed field,Eo(t)represents the pulse shape. The propagation velocity is given by v.Note that 1 =c兰2.998×103ms-1 (1.18) √u0e0 is the velocity of light in free space.Therefore,for the case most common in optics where u=uo,the velocity of propagation within a medium is given by v=C (1.19) n where n is the refractive index according to eq.(1.10).Note also that in deriving egs.(1.14) to(1.17),we have assumed implicitly that the refractive index n is independent of frequency. When n does have a frequency dependence,this can change the propagation velocity or cause the pulse to distort during propagation.These effects are discussed in Chapter 4. The case of a sinusoidal solution to the wave equation will be of special importance. Then eq.(1.17)takes the form E(z.t)=Eo cos(@t-kz+) (1.20) where Eo is now a constant vector,is the angular frequency,and the propagation constant k must satisfy the dispersion relation k =0us (1.21)
BRIEF REVIEW OF ELECTROMAGNETICS 7 Since ∇ · E = 0 also under our conditions, we obtain the wave equation ∇ 2E = µε ∂ 2E ∂t2 (1.15) One situation of special interest is the case where the field varies in only one direction, which without loss of generality we take as the z direction. Then the wave equation becomes ∂ 2E ∂z2 = µε ∂ 2E ∂t2 (1.16) The general solution takes the form E (z, t) = E0 t − z v (1.17) where E0 is a vector in the x–y plane [eq. (1.1) precludes E from having a z-component] and v = 1/ √ µε. The solution can be verified by plugging back into the wave equation. Equation (1.17) is called a plane-wave solution, since the field does not vary in the transverse (x– y) plane. It also represents a traveling wave, since the field propagates in the z direction without changing its form. In the case of a pulsed field, E0(t) represents the pulse shape. The propagation velocity is given by v. Note that 1 √ µ0ε0 = c ∼= 2.998 × 108 m s−1 (1.18) is the velocity of light in free space. Therefore, for the case most common in optics where µ = µ0, the velocity of propagation within a medium is given by v = c n (1.19) where n is the refractive index according to eq. (1.10). Note also that in deriving eqs. (1.14) to (1.17), we have assumed implicitly that the refractive index n is independent of frequency. When n does have a frequency dependence, this can change the propagation velocity or cause the pulse to distort during propagation. These effects are discussed in Chapter 4. The case of a sinusoidal solution to the wave equation will be of special importance. Then eq. (1.17) takes the form E (z, t) = E0 cos(ωt − kz + φ) (1.20) where E0 is now a constant vector, ω is the angular frequency, and the propagation constant k must satisfy the dispersion relation k = ω √ µε (1.21)
8 INTRODUCTION AND REVIEW or again,assuming that u=wo, k=咖 (1.22) The wave has a temporal oscillation period equal to 2/@and a spatial period or wavelength in the medium given by =2x/k.The wavelength in free space is denoted Ao and is given by 2πc 0= (1.23) Equation(1.20)represents the ideal case of single frequency or monochromatic laser radi- ation.It can also be written in the equivalent form E(z.t)=Re Eoej(c-kz) (1.24) where Ref...}denotes the real part and the phasehas been incorporated into the complex vector Eo.We refer to this form as complex notation.As we will see shortly,ultrashort light pulses are conveniently described as superpositions of sinusoidal solutions of the form (1.20)or (1.24)with different frequencies. Finally,we note that similar solutions can be written for propagation in directions other than along z,as follows: Er,=Re{色oeio-kr} (1.25) Here k is the propagation vector;it points along the direction of propagation and its mag- nitude k =k still satisfies the dispersion relation (1.21). 1.2.3 Poynting's Vector and Power Flow We also review the expressions for energy flow with electromagnetic waves.To arrive at the required formulas,we form the dot product of eq.(1.3)with H and subtract from this the dot product of eq.(1.4)with E.Using another vector identity,2 we find that aB aD VE×H)+H +E -+E.J=0 at (1.26 We also make use of the divergence theorem, V.Adv=A.nds (1.27) which states that the surface integral of a vector A over a closed surface is equal to the volume integral of V.A over the volume bounded by that surface.f is the unit vector 2 The identity is V.A x B=B.V x A-A.V x B
8 INTRODUCTION AND REVIEW or again, assuming that µ = µ0, k = ωn c (1.22) The wave has a temporal oscillation period equal to 2π/ω and a spatial period or wavelength in the medium given by λ = 2π/k. The wavelength in free space is denoted λ0 and is given by λ0 = 2πc ω (1.23) Equation (1.20) represents the ideal case of single frequency or monochromatic laser radiation. It can also be written in the equivalent form E(z, t) = Re E˜ 0e j(ωt−kz) (1.24) where Re{· · · } denotes the real part and the phase φ has been incorporated into the complex vector E˜ 0. We refer to this form as complex notation. As we will see shortly, ultrashort light pulses are conveniently described as superpositions of sinusoidal solutions of the form (1.20) or (1.24) with different frequencies. Finally, we note that similar solutions can be written for propagation in directions other than along z, as follows: E(r, t) = Re E˜ 0e j(ωt−k·r) (1.25) Here k is the propagation vector; it points along the direction of propagation and its magnitude k = |k| still satisfies the dispersion relation (1.21). 1.2.3 Poynting’s Vector and Power Flow We also review the expressions for energy flow with electromagnetic waves. To arrive at the required formulas, we form the dot product of eq. (1.3) with H and subtract from this the dot product of eq. (1.4) with E. Using another vector identity,2 we find that ∇ · (E × H) + H · ∂B ∂t + E · ∂D ∂t + E · J = 0 (1.26) We also make use of the divergence theorem, ∇ · A dV = A · nˆ dS (1.27) which states that the surface integral of a vector A over a closed surface is equal to the volume integral of ∇ · A over the volume bounded by that surface. nˆ is the unit vector 2 The identity is ∇ · A × B = B · ∇ × A − A · ∇ × B
BRIEF REVIEW OF ELECTROMAGNETICS 9 normal to the surface and pointing outward.The result is ∫B×面a+w{红+E+EJ}-0 (1.28) Finally,assuming a linear medium and substituting for D and B using eqs.(1.8)and(1.12), we obtain (E×H)·idS+ ) at =0 (1.29) Equations (1.28)and (1.29)are representations of Poynting's theorem,which describes conservation of energy in electromagnetic systems.We can identify specific meanings for each of the terms.Look at eq.(1.29),for example: ·∫(E×H)·nds is the net rate of energy flow out of the closed surface.It has units of power(watts).Ex H is called the Poynting vector and has units of intensity (W/m2). It gives the power density carried by an electromagnetic wave and the direction in which power is carried. .E2andare the local energy densities (J/m3)associated with the electric and magnetic fields,respectively.a/atdV and a/atudV represent the time rate of change of electric and magnetic field energy stored within the volume, respectively. .fE.J dV represents power dissipation or generation within the volume(in watts). When E.J is positive,this term represents power dissipation due,for example,to ohmic losses.Energy is transferred out of the fields and into the medium,typically as heat.When E.J is negative,this term represents power supplied by the currents and fed into the electromagnetic fields. Overall,Poynting's theorem is a power balance equation,showing how changes in stored energy are accounted for by power dissipation and energy flow. It is worth specializing once more to the case of single-frequency sinusoidal fields,with E given by eq.(1.25).The H field is obtained using eq.(1.3),with the result H=Re Ek×Eoca-kr) (1.30) Thus H is perpendicular to both E and k,and its magnitude is equal to/.The factor u/s is termed the characteristic impedance of the medium,and velu is therefore the admittance
BRIEF REVIEW OF ELECTROMAGNETICS 9 normal to the surface and pointing outward. The result is (E × H) · nˆ dS + dV H · ∂B ∂t + E · ∂D ∂t + E · J = 0 (1.28) Finally, assuming a linear medium and substituting for D and B using eqs. (1.8) and (1.12), we obtain (E × H) · nˆ dS + dV ∂ 1 2 ε |E| 2 ∂t + ∂ 1 2 µ|H| 2 ∂t + E · J = 0 (1.29) Equations (1.28) and (1.29) are representations of Poynting’s theorem, which describes conservation of energy in electromagnetic systems. We can identify specific meanings for each of the terms. Look at eq. (1.29), for example: (E × H) · nˆ dS is the net rate of energy flow out of the closed surface. It has units of power (watts). E × H is called the Poynting vector and has units of intensity (W/m2 ). It gives the power density carried by an electromagnetic wave and the direction in which power is carried. 1 2 ε |E| 2 and 1 2 µ|H| 2 are the local energy densities (J/m3 ) associated with the electric and magnetic fields, respectively. ∂/∂t 1 2 ε |E| 2 dV and ∂/∂t 1 2 µ|H| 2 dV represent the time rate of change of electric and magnetic field energy stored within the volume, respectively. E · J dV represents power dissipation or generation within the volume (in watts). When E · J is positive, this term represents power dissipation due, for example, to ohmic losses. Energy is transferred out of the fields and into the medium, typically as heat. When E · J is negative, this term represents power supplied by the currents and fed into the electromagnetic fields. Overall, Poynting’s theorem is a power balance equation, showing how changes in stored energy are accounted for by power dissipation and energy flow. It is worth specializing once more to the case of single-frequency sinusoidal fields, with E given by eq. (1.25). The H field is obtained using eq. (1.3), with the result H = Re ε µ k × E˜ 0 k e j(ωt−k·r) (1.30) Thus H is perpendicular to both E and k, and its magnitude is equal to √ ε/µ|E|. The factor √ µ/ε is termed the characteristic impedance of the medium, and √ ε/µ is therefore the admittance.