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K. E. Oughstun and R. A. Albanese Vol. 23, No. 7 / July 2006 / J. Opt. Soc. Am. A 1751 Magnetic field contribution to the Lorentz model Kurt E. Oughstun College of Engineering and Mathematical Sciences, University of Vermont, Burlington, Vermont 05405-0156 Richard A. Albanese Information Operations and Special Projects Division, U. S. Air Force Research Laboratory, Brooks City-Base, Texas 78235 Received November 10, 2005; accepted December 8, 2005; posted January 24, 2006 (Doc. ID 65838) The classical Lorentz model of dielectric dispersion is based on the microscopic Lorentz force relation and Newton’s second law of motion for an ensemble of harmonically bound electrons. The magnetic field contribution in the Lorentz force relation is neglected because it is typically small in comparison with the electric field contribution. Inclusion of this term leads to a microscopic polarization density that contains both perpendicular and parallel components relative to the plane wave propagation vector. The modified parallel and perpendicular polarizabilities are both nonlinear in the local electric field strength. © 2006 Optical Society of America OCIS codes: 260.2030, 190.4400, 160.4760, 160.4330. 1. INTRODUCTION 1–3 The Lorentz model of resonant dispersion phenomena in dielectric materials is a classical model of central importance in optics4,5 as well as in the broader discipline of electromagnetics.6 It is a causal model6,7 that describes both normal and anomalous dispersion phenomena from the infrared through the optical regions of the electromagnetic spectrum. This model is based on the microscopic Lorentz force relation f共r,t兲 = 共r,t兲e共r,t兲 + 1 储c储 j共r,t兲 ⫻ b共r,t兲, 共1兲 where 共r , t兲 is the microscopic change density and j共r , t兲 = 共r , t兲dr / dt the microscopic current density of the charged particle at the space–time point 共r , t兲 and where f共r , t兲 is the microscopic force density exerted on the charged particle by the electromagnetic field with microscopic electric intensity vector e共r , t兲 and magnetic induction vector b共r , t兲. The quantity ⴱ appearing in the double brackets 储ⴱ储 in any equation here is used as a conversion factor between cgs and MKS units.6,8 If that factor is included in that equation, then the equation is in cgs units, while if the factor is omitted from that equation, then it is in MKS units. If no such factor appears in a given equation, then that equation is correct in either system of units. The magnetic field contribution appearing on the righthand side of Eq. (1) is typically assumed to be negligible in comparison with the electric field contribution for sufficiently small field strengths and values of the relative velocity v / c of the charged particle, where v = 兩dr / dt兩. This assumption then leads to the classical Lorentz theory of dielectric dispersion in which electric field effects alone are considered. The question then remains as to what effects are introduced by the inclusion of the magnetic field contribution in the Lorentz theory. The solution of this important problem has been partially addressed9–11 for 1084-7529/06/071751-6/$15.00 the special case of a chiral molecule, where it has been shown11 that the magnitude of the magnetic response of the induced second-order optical activity can be comparable to the magnitude of the electric response alone. However, as we have been unable to find any previously published general solution of this problem, it is addressed in this paper. 2. MODIFIED LORENTZ MODEL OF DIELECTRIC DISPERSION With the complete Lorentz force relation given in Eq. (1) as the driving force, the equation of motion of a harmonically bound electron is given by d 2r j dt 2 + 2␦j drj dt + j2rj = − qe m 冉 Eeff共r,t兲 + 1 drj 储c储 dt 冊 ⫻ Beff共r,t兲 , 共2兲 where Eeff共r , t兲 is the effective local electric field intensity and Beff共r , t兲 is the effective local magnetic induction field at the space–time point 共r , t兲 and where rj = rj共r , t兲 describes the displacement of the electron from its equilibrium position. Here qe denotes the magnitude of the charge and m the mass of the harmonically bound electron with undamped resonance frequency j and phenomenological damping constant ␦j. The temporal Fourier integral representation of the electric and magnetic field vectors of the effective local plane electromagnetic wave is given by8 Ẽeff共r, 兲 = 冕 ⬁ −⬁ © 2006 Optical Society of America Eeff共r,t兲eitdt, 共3兲 1752 J. Opt. Soc. Am. A / Vol. 23, No. 7 / July 2006 B̃eff共r, 兲 = 冕 K. E. Oughstun and R. A. Albanese ⬁ Beff共r,t兲eitdt r̃j · k = − i −⬁ = 储c储 k共兲 ⫻ Eeff共r, 兲, 共4兲 where k共兲 is the wavevector of the plane wave field with magnitude given by the wave number k共兲 = / c in vacuum, since the effective local field is essentially a microscopic field. With the temporal Fourier integral representation r̃j共兲 = 冕 the dynamical equation of motion given in Eq. (2) becomes 共2 + 2i␦j − j2兲r̃j = m qe = m r̃j · Ẽeff = qe/m − j2 + 2i␦j 2 冋 ⫻ 1+ 2 2 共qe/mc兲2Ẽeff 2 共2 − j2 + 2i␦j兲2 − 共qe/mc兲2Ẽeff 2 qe 冋 aj = qe/m − j2 + 2i␦j 2 冋 ⫻ 1+ 关共1 + ir̃j · k兲Ẽeff − i共r̃j · Ẽeff兲k兴, bj = − i 1 + ir̃j · k m − 2 j2 + 2i␦j Ẽeff − i r̃j · Ẽeff − 2 j2 + 2i␦j 册 k . 共7兲 The electron displacement vector may then be expressed as a linear combination of the orthogonal pair of vectors k and Ẽeff as 共8兲 r̃j = ajẼeff + bjk, where, because of the transversality relation k · Ẽeff = 0, aj = r̃j · Ẽeff 2 Ẽeff 2 2 共qe/mc兲2Ẽeff 2 共2 − j2 + 2i␦j兲2 − 共qe/mc兲2Ẽeff 2 2 共qe/mc兲2Ẽeff 2 共2 − j2 + 2i␦j兲2 − 共qe/mc兲2Ẽeff 2 , 册 , 共13兲 共14兲 respectively. The local (or microscopic) induced dipole moment p̃j ⬅ −qer̃j for the jth Lorentz oscillator type is then given by p̃j共r, 兲 = − qe共ajẼeff共r, 兲 + bjk兲. 共15兲 If there are Nj Lorentz oscillators per unit volume of the jth type, then the macroscopic polarization induced in the medium is given by the summation over all oscillator types of the spatially averaged locally induced dipole moments as P̃共r, ,Ẽeff兲 = 兺 N 具具p̃ 共r, 兲典典 j j j 共9兲 , 2 Ẽeff . The coefficients aj and bj appearing in Eq. (8) are then given by 共6兲 r̃j = 册 共12兲 共Ẽeff − ir̃j ⫻ 共k ⫻ Ẽeff兲兲 with formal solution 共11兲 , 共5兲 −⬁ qe 2 共2 − j2 + 2i␦j兲2 − 共qe/mc兲2Ẽeff 2 and substitution of this result into the second relation gives ⬁ rj共t兲eitdt, 2 共qe/mc兲2Ẽeff 2 = 具具Ẽeff共r, 兲典典 兺N␣ j j⬜共,Ẽeff兲 j c2 bj = r̃ 2 j +k 共10兲 · k. 兺 N ␣ 共,Ẽ j j储 eff兲. 共16兲 j Here The pair of scalar products appearing in the above expression may be evaluated using the expression given Eq. (7) as 共0兲 共2兲 ␣j⬜共,Ẽeff兲 ⬅ ␣j⬜ 共兲 + ␣j⬜ 共,Ẽeff兲 = r̃j · k = − i r̃j · Ẽeff = qe qe r̃j · Ẽeff 2 m 2 − j2 + 2i␦j 1 + ir̃j · k m 2 − j2 + 2i␦j k , 2 Ẽeff . Substitution of the second relation into the first then yields − q2e /m 2 − j2 + 2i␦j 冋 ⫻ 1+ 2 2 共qe/mc兲2Ẽeff 2 共2 − j2 + 2i␦j兲2 − 共qe/mc兲2Ẽeff 2 册 共17兲 is defined as the perpendicular component of the atomic 共0兲 polarizability, with ␣j⬜ 共兲 ⬅ ␣j⬜共 , 0兲 = ␣j共兲, where ␣j共兲 is the classical expression for the atomic polarizability K. E. Oughstun and R. A. Albanese Vol. 23, No. 7 / July 2006 / J. Opt. Soc. Am. A when magnetic field effects are neglected, and ␣j储共,Ẽeff兲 ⬅ i 2 共q3e /m2兲Ẽeff 2 共2 − j2 + 2i␦j兲2 − 共qe/mc兲2Ẽeff 2 共18兲 is defined here as the parallel component of the atomic polarizability, where ␣j储共 , 0兲 = 0. 3. CRITICAL FIELD STRENGTH The atomic polarizability is then seen to be nonlinear in the local electric field strength when magnetic field effects 共2兲 共 , Ẽeff兲 appearing in are included. The nonlinear term ␣j⬜ Eq. (17) will be negligible in comparison with the linear 共0兲 term ␣j⬜ 共兲 when the local electric field strength is sufficiently smaller than the critical field strength Ecrit = Ecrit共兲 defined by the condition 冏 2 共qe/mc兲2Ẽcrit 2 2 共2 − j2 + 2i␦j兲2 − 共qe/mc兲2Ẽcrit 2 冏 1753 undefined; the estimate given by Eq. (21) yields an approximate value of 7.5⫻ 1011 V / m. Notice that Ecrit → ⬁ at both of the near-resonance values 0± as well as when either → 0 or → ⬁. Notice also that since megavolt per meter electric field strengths are produced by ultrashort pulsed terawatt laser systems, the minimum critical field strength for this example is at least five orders of magnitude greater than that which is currently available. The angular frequency dependence of the relative cor共2兲 共0兲 rection factor ␣j⬜ 共 , Ẽeff兲 / ␣j⬜ 共兲 for the perpendicular atomic polarizability given in Eq. (17) is illustrated in Fig. 3 for the near-infrared resonance line of triply distilled water for several values of the relative local electric field intensity about unity. Notice that this correction factor reaches its peak value at the resonance frequency 0 when 兩Ẽeff / 共Ecrit兲min兩 ⬍ 1 and that this peak value bifurcates into a pair of symmetric peaks about this resonance frequency when 兩Ẽeff / 共Ecrit兲min兩 ⬎ 1, a local minimum now ⬅ 1, with solution Ecrit共兲 = mc 共2 − j2兲2 + 4␦j22 冑2qe 关共2 − j2兲2 − 4␦j22兴1/2 . 共19兲 When 兩Ẽeff兩 Ⰶ Ecrit, the linear term in Eq. (17) dominates 共0兲 共兲. Notice the nonlinear term so that ␣j⬜共 , Ẽeff兲 ⯝ ␣j⬜ that this critical electric field strength depends on the value of the applied angular frequency of the electromagnetic wave field. Furthermore, notice that this critical field strength in undefined in the angular frequency band 苸 关j− , j+兴, where j± ⬅ 共j2 + 2␦j2 ± 2␦j冑j2 + ␦j2兲1/2 共20兲 j 苸 关j− , j+兴. for each resonance feature j, where Finally, a rough estimate of the minimum value for this critical field strength is seen to be given by 共Ecrit兲min ⬇ mc 冑2qe j = បj 冑8 B 共21兲 Fig. 1. Real (solid curve) and imaginary (dashed curve) parts of the linear atomic polarizability for the near-infrared resonance line in water. for each resonance feature, where B ⬅ qeប / 2mc is the Bohr magneton. The quantity បj is recognized as the energy level state of the harmonically bound electron. 4. NUMERICAL EXAMPLE Consider a single resonance Lorentz model dielectric 共j = 0兲 with material parameters representative of the nearinfrared line in triply distilled water at 25 ° C, where 0 = 6.19⫻ 1014 r / s and ␦0 = 2.86⫻ 1013 r / s. The angular frequency dependence of the real and imaginary parts of the 共0兲 linear atomic polarizability ␣j⬜ ⬅ ␣j共兲 for this single resonance line is presented in Fig. 1, and the angular frequency dependence of the critical field strength described by Eq. (19) is illustrated in Fig. 2, where 0− ⯝ 5.91 ⫻ 1014 r / s and 0+ ⯝ 6.48⫻ 1014 r / s. The minimum critical electric field strength 共Ecrit兲min ⯝ 1.38⫻ 1011 V / m is seen to occur at the two points just below and above the frequency band 关0− , 0+兴 where the critical field strength is Fig. 2. Angular frequency dependence of the critical local electric field strength (in volts per meter) for the near-infrared resonance line in water. 1754 J. Opt. Soc. Am. A / Vol. 23, No. 7 / July 2006 Fig. 3. Angular frequency dependence of the correction factor for the perpendicular atomic polarizability for several values of the relative local electric field strength. K. E. Oughstun and R. A. Albanese electric field strength goes to zero and so represents the classical Lorentz result when magnetic field effects are neglected. Magnetic field effects are entirely negligible until the minimum critical field strength is approached from below. At the critical field strength the real transverse atomic polarizability has just started to bifurcate into a pair of resonance peaks, one below the linear resonance frequency 0 and the other above that characteristic angular frequency. Notice that the downshifted nonlinear resonance is stronger than the upshifted nonlinear resonance. The downshifted nonlinear resonance peak increases as the local field strength continues to increase above the minimum critical field strength while the upshifted peak continues to decrease in strength. Similar results are obtained for the imaginary part of the transverse atomic polarizability, illustrated in Fig. 5. An approximate expression for the locations of these upshifted and downshifted nonlinear resonance peaks may be obtained from the appropriate zeros of the denominator in the correction factor appearing in Eq. (17). The zeros of this expression are given by the roots of the quadratic equation 2 − j2 + 2i␦j = ± 共qe / mc兲Ẽeff. Neglect of the damping term results in the approximate pair of positive frequency solutions ± ⬇ BẼeff ប 再 冋 冉 冊册 冎 ±1 + 1 + បj BẼeff 2 1/2 , 共22兲 where lim ± ⬇ j . 共23兲 ˜ →0 E eff In the opposite limit when Ẽeff Ⰷ បj / B, the limiting values Fig. 4. Angular frequency dependence of the real part of the transverse atomic polarizability for critical and supercritical values of the local electric field strength compared to the linear behavior at zero field strength. appearing at the resonance frequency 0. As this relative local electric field strength continues to increase above unity, these two peak values continue to separate as they also increase in strength. The minimum value at 0, however, becomes nearly constant so that the curves cross each other as the frequency is either increased to the upper peak or decreased to the lower peak from this minimum value point. Finally, since this correction factor is below unity for all real frequencies when 兩Ẽeff / 共Ecrit兲min兩 ⬍ 1, magnetic field effects can be safely neglected in the Lorentz model of the perpendicular atomic polarizability when this inequality is well satisfied. The nonlinear character of the real part of the perpendicular atomic polarizability that is described by Eq. (17) is presented in Fig. 4 for local electric field strength values increasing above the minimum critical field strength. The zero field curve in the figure describes the linear atomic polarizability obtained in the limit as the local Fig. 5. Angular frequency dependence of the imaginary part of the transverse atomic polarizability for critical and supercritical values of the local electric field strength compared with the linear behavior at zero field strength. K. E. Oughstun and R. A. Albanese Vol. 23, No. 7 / July 2006 / J. Opt. Soc. Am. A 1755 resonance about the undamped resonance frequency 0 when the local electric field strength is equal to or less than the minimum critical field strength 共Ecrit兲min given in Eq. (21). However, as the local field strength exceeds this minimum critical value, this single resonance peak is found to split into a pair of resonance structures, one downshifted and the other uphifted in angular frequency from the characteristic resonance frequency 0 of the Lorentz model dielectric. 5. DISCUSSION Fig. 6. Angular frequency dependence of the real part of the longitudinal atomic polarizability for critical and subcritical values of the local electric field strength. Fig. 7. Angular frequency dependence of the imaginary part of the longitudinal atomic polarizability for critical and subcritical values of the local electric field strength. + ⬇ បj2 2BẼeff − ⬇ + 2B ប បj2 2BẼeff , Ẽeff , The results presented here show the precise manner in which the magnetic field influences the atomic polarizability in the Lorentz model of resonance polarization. Inclusion of the magnetic field in the Lorentz force relation results in a microscopic polarization density that contains both perpendicular and parallel components relative to the plane wave propagation vector of the local driving field that are both nonlinear in the local electrical field strength. This nonlinearity becomes significant when the local applied electric field strength exceeds a minimum critical field strength whose value increases linearly with the resonance frequency. Numerical calculations show that these nonlinear terms are entirely negligible for effective field strengths that are typically less than ⬃1012 V / m and that they begin to have a significant contribution for field strengths that are typically greater than ⬃1015 V / m for a highly absorptive material. Fortunately, this critical field strength is at least five orders of magnitude greater than that which is currently available in ultrashort pulsed terawatt laser systems. Although this magnetic field effect is negligible in most practical situations, it has been shown to be significant in materials exhibiting chirality,9–11 and it may also become important for artificial materials. Nevertheless, seemingly excessive electric field strengths commonly occur in nature; for example, it is estimated12 that the critical field strength for the alignment of water molecules for crystallization into polar ice crystals is greater than 109 V / m. 共24兲 共25兲 are obtained. Similar results are obtained for the frequency dependence of the nonlinear character of the parallel atomic polarizability given in Eq. (18). Although this parallel component of the atomic polarizability vanishes as Ẽeff → 0, it can exceed the strength of the transverse component of the atomic polarizability when Ẽeff exceeds the minimum critical field strength. The frequency dependence of the real part of the longitudinal (parallel) component is presented in Fig. 6 and the imaginary part in Fig. 7 for two values of the local electric field strength. Both the real and the imaginary parts are seen to exhibit a pronounced Fig. 8. Angular frequency dependence of the relative phasor velocity magnitude of the harmonically bound electron in the Lorentz model. 1756 J. Opt. Soc. Am. A / Vol. 23, No. 7 / July 2006 K. E. Oughstun and R. A. Albanese Notice that relativistic effects need not be considered in this extended model when the local field strength is equal to or below 10% of this minimum critical field strength since the magnitude of the phasor velocity v ⬅ r̃˙ = −ir̃ j j j for the harmonically bound electron, where r̃j is given by Eq. (8), remains below ⬃0.02c when Ẽeff / 共Emin兲min 艋 0.1 for the example considered here, the maximum value occurring near resonance. However, relativistic effects must be considered when the minimum critical field strength is reached and exceeded, especially near resonance, as illustrated in Fig. 8. The physical origin of the nonlinear term considered here is due to the diamagnetic effect that appears in the analysis of the interaction of an electromagnetic field with a charged particle in the quantum theory of electrodynamics.13,14 The Hamiltonian for this coupled system is given by (see Eqs. XIII.71–XIII.72 of Messiah13) H = H0 − qe 2mc H·L+ q2e 8mc2 H 兺 ACKNOWLEDGMENT The research presented in this paper has been supported by the U.S. Air Force Office of Scientific Research under AFOSR grants F49620-01-0306 and USAF 9550-04-10447. The authors can be reached by e-mail at [email protected] and [email protected]. REFERENCES 1. 2. Z 2 taken to be given by the Bohr radius a0 ⬅ ប2 / mq2e ⬇ 5.29 ⫻ 10−9 cm, where ប ⬅ h / 2 and h is Planck’s constant, the electric field strength is E ⬇ 5.13⫻ 1011 V / m, in general agreement with the preceeding estimate of the minimum critical field strength. 2 rj⬜ , j=1 in Gaussian units, where H0 is the Hamiltonian of the center of mass system of the isolated atom with Z spinless electrons, H共r兲 is the magnetic field intensity vector with magnitude H ⬅ 兩H兩, where r⬜ is the projection of the position vector r on the plane perpendicular to H共r兲, and Z 共rj ⫻ pj兲 is the total angular momentum of where L = 兺j=1 the Z atomic electrons. The third term in the above expression for the Hamiltonian is the main factor in atomic diamagnetism. The order of magnitude of this factor is given by ⬃共Zq2e / 12mc2兲H2具r2典, where 具r2典 ⬃ 1 ⫻ 10−16 cm2 for a bound electron. The ratio of this quantity to the level distance BH, where B ⬅ qeប / 2mc is the Bohr magneton, is found13 to be ⬃10−9ZH Gauss. for a single electron atom 共Z = 1兲, the diamagnetic effect will become significant when H 艌 109 Gauss, which corresponds to an electric field strength E 艌 109 esu or, equivalently, E 艌 3 ⫻ 1013 V / m, in agreement with the preceding classical result that the nonlinear effects in the Lorentz model become significant for an applied field strength between 1012 V / m and 1015 V / m. In addition, nonlinear optical effects are found to dominate the linear response when the local field strength becomes comparable to the Coulomb field of the atomic nucleus.15 As an estimate of this field strength, if the distance between the nucleus and the bound electron is 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. H. A. Lorentz, Versuch einer Theorie der Electrischen und Optischen Erscheinungen in Bewegten Körpern (Teubner, 1906). H. A. Lorentz, The Theory of Electrons (Dover, 1952), Chap. 4. L. Rosenfeld, The Theory of Electrons (Dover, 1965). J. M. Stone, Radiation and Optics (McGraw-Hill, 1963), Chap. 15. M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 1999), Chap. 2. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999), Chap. 7. H. M. Nussenzveig, Causality and Dispersion Relations (Academic, 1972), Chap. 1. K. E. Oughstun and G. C. Sherman, Electromagnetic Pulse Propagation in Causal Dielectrics (Springer-Verlag, 1994), Chap. 1. M. Kauranen, T. Verbiest, and A. Persoons, “Electric and magnetic contributions to the second-order optical activity of chiral surfaces,” Mol. Cryst. Liq. Cryst. Sci. Technol., Sect. B: Nonlinear Opt. 8, 243–249 (1994). M. Kauranen, T. Verbiest, E. W. Meijer, E. E. Havinga, M. N. Teerenstra, A. J. Schouten, R. J. M. Nolte, and A. Persoons, “Chiral effects in the second-order optical nonlinearity of a poly(isocyanide) monolayer,” Adv. Mater. (Weinheim, Ger.) 7, 641–644 (1995). J. J. Maki and A. Persoons, “One-electron second-order optical activity of a helix,” J. Chem. Phys. 104, 9340–9348 (1996). E.-M. Choi, Y.-H. Yoon, S. Lee, and H. Kang, “Freezing transition of interfacial water at room temperature under electric fields,” Phys. Rev. Lett. 95, 085701 (2005). A. Messiah, Quantum Mechanics (North-Holland, 1962), Vol. II. C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms: Introduction to Quantum Electrodynamics (Wiley, 1989), Subsect. III.D. R. W. Boyd, Nonlinear Optics (Academic, 1992), Chap. 1.