Nonlinear Optics - Appendices doc

Appendices Appendix A. The SI System of Units In this appendix we review briefly the basic equations of electromagnetism when written in the SI (System International, or rationalized mksa) system of units. Conversion between the SI and gaussian systems of units is summarized in an additional appendix. The intent of this appendix is to establish notation and not to present a rigorous exposition of electromagnetic theory. In the SI system, mechanical properties are measured in mks units, that is, distance is measured in meters (m), mass in kilograms (kg), and time in seconds (s). The unit of force is thus the kg m/sec 2 , known as a newton (N), and the unit of energy is the kg m 2 /sec 2 , known as the joule (J). The fundamental electrical unit is a unit of charge, known as the coulomb (C). It is defined such that the force between two charged point particles, each containing 1 coulomb of charge and separated by a distance of 1 meter, is 1 newton. More generally, the force between two charged particles of charges q 1 and q 2 separated by the directed distance r =r ˆ r, where ˆ r is a unit vector in the r direction, is given by F = q 1 q 2 4π 0 r 2 ˆ r. (A.1) This result is known as Coulomb’s law. The parameter  0 that appears in this equation is known as the permittivity of free space and has the value  0 = 8.85 × 10 −12 F/m. Here F is the abbreviation for the farad, which is defined as 1 coulomb/volt. The unit of electrical current is the ampere (A), which is 1 coulomb/sec. The unit of electrical potential (i.e., potential energy per unit charge) is the volt, which is 1 joule/coulomb. In the SI system, Maxwell’s equations have the form ∗ ∇×E =− ∂B ∂t , (A.2a) ∇×H = ∂D ∂t +J, (A.2b) ∗ In this appendix, we dispense with our usual notation of using a tilde to denote time-varying quantities. 589 590 Appendices ∇·D = ρ, (A.2c) ∇·B = 0. (A.2d) The units and names of the field vectors are as follows: [E]=V/m (electric field), (A.3a) [D]=C/m 2 (electric displacement), (A.3b) [B]=T (magnetic field, or magnetic induction), (A.3c) [H]=A/m (magnetic intensity), (A.3d) [P]=C/m 2 (polarization), (A.3e) [M]=A/m (magnetization). (A.3f) In Eq. (A.3c), T notes the tesla, the unit of magnetic field strength. The tesla is equivalent to the Wb /m 2 , where Wb denotes the weber, the unit of magnetic flux, which is equivalent to 1 joule/ampere or to 1 volt second. The vectors P and M are known as the polarization and magnetization, respectively. The polarization P represents the electric dipole moment per unit volume that may be present in a material. The magnetization M denotes the magnetic dipole moment per unit volume that may be present in the material. These quantities are discussed further in the discussion given below. The two additional quantities appearing in Maxwell’s equations are the free charge density ρ, measured in units of coulombs/m 3 , and the free current density J, measured in units of amperes/m 2 . Under many circumstances, J is given by the expression J = σ E, (A.4) which can be considered to be a microscopic form of Ohm’s law. Here σ is the electrical conductivity, whose units are ohm −1 m −1 . The ohm is the unit of electrical resistance and has units of volt/ampere. The relationships that exist among the f our electromagnetic field vectors because of purely material properties are known as the constitutive relations. These relations, even in the presence of nonlinearities, have the form D =  0 E +P, (A.5a) H = μ −1 0 B −M. (A.5b) Here μ 0 is the magnetic permeability of free space, which has the value μ 0 = 1.26 × 10 −6 H/m. Here H is the abbreviation for the henry, which is defined as 1 weber/ampere or as 1 volt second/ampere. Appendix A. The SI System of Units 591 The manner in which the response of a material medium can lead to a nonlinear dependence of P upon E is of course the subject of this book. For the limiting case of a purely linear response, the relationships can be expressed (assuming an isotropic medium for notational simplicity) as P =  0 χ (1) E, (A.6a) M = χ (1) m H. (A.6b) Note that the linear electric susceptibility χ (1) and the linear magnetic susceptibility χ (1) m are dimensionless quantities. We now introduce the linear relative dielectric constant  (1) and the linear relative magnetic permeability μ (1) , both of which are dimensionless and are defined by D =  0  (1) E, (A.7a) B = μ 0 μ (1) m H. (A.7b) We then find by consistency of Eqs. (A.5a), (A.6a), and (A.7a) and of (A.5b), (A.6b), and (A.7b) that  (1) = 1 +χ (1) , (A.8a) μ (1) = 1 +χ (1) m . (A.8b) The fields E and B (rather than D and H) are usually taken to constitute the fundamental electromagnetic fields. For example, the force on a particle of charge q moving at velocity v through an electromagnetic field is given by the Lorentz force law in the form F = q  E +(v ×B)  . (A.9) A.1. Energy Relations and Poynting’s Theorem Poynting’s theorem can be derived from Maxwell’s equations in the following manner. We begin with the vector identity ∇·(E ×H) =H · (∇×E) −E ·(∇×H) (A.10) and introduce expressions for ∇×E and ∇×H from the Maxwell equations (A.2a) and (A.2b) to obtain ∇·(E ×H) +  H · ∂B ∂t +E · ∂D ∂t  =−J ·E. (A.11) 592 Appendices Assuming for simplicity the case of a purely linear response, the second term on the left-hand side of this equation can be expressed as ∂u/∂t, where u = 1 2 (E ·D +B · H) (A.12) represents the energy density of the electromagnetic field. We also introduce the Poynting vector S = E ×H, (A.13) which gives the rate at which electromagnetic energy passes through a unit area whose normal is in the direction of S. Equation (A.11) can then be written as ∇·S + ∂u ∂t =−J ·E, (A.14) where J · E gives the rate per unit volume at which energy is lost to the field through Joule heating. A.2. The Wave Equation A wave equation for the electric field can be derived from Maxwell’s equations, as described in greater detail in Section 2.1. We assume the case of a linear, isotropic, nonmagnetic (i.e., μ =1) medium that is free of sources (i.e., ρ =0andJ = 0). We first take the curl of the first Maxwell equation (A.2a), reverse the order of differentiation on the right-hand side, replace B by μ 0 H, and use the second Maxwell equation (A.2b) to replace ∇×H by ∂D/∂t to obtain ∇×∇×E =−μ 0 ∂ 2 D ∂t 2 . (A.15) On the left-hand side of this equation, we make use of the vector identity ∇×∇×E =∇(∇·E) −∇ 2 E, (A.16) and drop the first term because ∇·E must vanish whenever ρ vanishes in an isotropic medium because of the Maxwell equation (A.2c). On the right-hand side, we replace D by  0  (1) E, and set μ 0  0 equal to 1/c 2 . We thus obtain the wave equation in the form −∇ 2 E +  (1) c 2 ∂ 2 E ∂t 2 =0. (A.17) This equation possesses solutions in the form of infinite plane waves—that is, E = E 0 e i(k·r−ωt) +c.c., (A.18) Appendix A. The SI System of Units 593 where k and ω must be related by k = nω/c where n =   (1) and k =|k|. The magnetic intensity associated with this wave has the form H = H 0 e i(k·r−ωt) +c.c. (A.19) Note that, in accordance with the convention followed in this book, factors of 1 2 are not included in these expressions. From Maxwell’s equations, one can deduce that E 0 , H 0 ,andk are mutually orthogonal and that the magnitudes of E 0 and H 0 are related by n|E 0 |=  μ 0 / 0 |H 0 |. (A.20) The quantity √ μ 0 / 0 is known as the impedance of free space and has the value 377 ohms. Since  0 μ 0 =1/c 2 , the impedance of free space can alterna- tively be written as √ μ 0 / 0 =1/ 0 c. In considerations of the energy relations associated with a time-varying field, it is useful to introduce a time-averaged Poynting vector S and a time-averaged energy density u. Through use of Eqs. (A.18)–(A.20) and the defining relations (A.12) and (A.13), we find that these quantities are given by S=2n   0 /μ 0 |E 0 | 2 ˆ k = 2n 0 c|E 0 | 2 ˆ k, (A.21a) u=2n 2  0 |E 0 | 2 , (A.21b) where ˆ k is a unit vector in the k direction. In this book the magnitude of the time-averaged Poynting vector is called the intensity I =|S| and is given by I =2n   0 /μ 0 |E 0 | 2 =2n 0 c|E 0 | 2 . (A.22) A.3. Boundary Conditions There are many situations in electromagnetic theory in which one needs to calculate the fields in the vicinity of a boundary between two regions of space with different optical properties. The way in which the fields are related on the opposite sides of the boundary constitutes the topic of boundary conditions. To treat this topic, we first express the Maxwell equations in their integral rather than differential forms. We recall the divergence theorem, which states that, for any well-behaved vector field A, the following identity holds:  V ∇·A dV =  S A ·n da. (A.23) 594 Appendices The integral on the left-hand side is to be performed over a closed three- dimensional volume V and the integral on the right-hand side is to be performed over the surface S that encloses this volume. The quantity n represents a unit vector pointing in the outward normal direction. If the divergence theorem is applied to Maxwell’s equations (A.2c) and (A.2d), one obtains  S D ·n da =  V ρdV, (A.24)  S B ·n da =0. (A.25) The first of these equations expresses Gauss’s law, and the second the absence of magnetic monopoles. We can similarly express the two “curl” Maxwell equations in integral form through use of Stokes’s theorem, which states that for any well-behaved vector field A  S (∇×A) ·n da =  C A ·dl. (A.26) Here S is any open surface, C is a curve that bounds it, and dl is a directed line element along this curve. When this theorem is applied to Maxwell’s equations (A.2a) and (A.2b), one obtains  C E ·dl =−  S ∂B ∂t ·n da, (A.27)  C H ·dl =  S  ∂D ∂t +J  ·n da. (A.28) The first of these equations expresses Faraday’s law, and the second expresses Ampere’s law with the inclusion of Maxwell’s displacement current. We are now in a position to determine the nature of the boundary conditions on the electromagnetic fields. We refer to Fig. A.1, which shows the interface between regions 1 and 2. We first imagine placing a small cylindrical pill box near the interface so that one circular side extends into region 1 and the other into region 2. We apply Eq. (A.24) to this situation. We next imagine shrinking the height of the pill box while keeping the areas of the two surfaces fixed. By such a limiting procedure, we are assured that the value of the surface integral is dominated by the fields on the two circular surfaces. We further assume that the pill box is sufficiently small that the fields are essentially constant over these surfaces. Even though the surface integrals then remain appreciable, the volume integral will vanish so long as ρ remains finite, because the volume over which the integration is performed will tend to zero as the height of Appendix A. The SI System of Units 595 FIGURE A.1 Constructions used to determine the boundary conditions of the electromagnetic fields at the interface (surface S) between regions 1 and 2. the pill box is shrunk. The only situation in which the volume integral can be nonvanishing is that in which ρ diverges somewhere within the region of integration, for example, if there is charge located on the surface separating regions 1 and 2. If we let  denote the surface charge density—that is, the charge per unit area located on the surface, we find that the boundary condition on D is given by (D 2 −D 1 ) ·n =. (A.29) The boundary condition on B is found much more simply. Since the right- hand side of Eq. A.25 vanishes, we find immediately that (B 2 −B 1 ) ·n =0. (A.30) Equation (A.30) tells us that the normal component of the B field must be continuous at the boundary. Equation (A.29) tells us that the normal component of the D field can be discontinuous but only by an amount equal to the charge density accumulated on the surface. This free-charge density can be appreciable for the case of metallic surfaces. However, the surface charge density vanishes at the interface between two dielectric materials. The boundary conditions for E and H can be determined by considering the path integral shown at right-hand side of the figure. We assume that the long sides of the path lie parallel to the surface, one on each side of the interface. We further assume the limiting situation in which the short sides and very much shorter than the long sides. In this situation the line integrals are dominated by the long sides of the paths, and the surface integrals tend to vanish because the area of the region of integration tends to zero. The surface integrals of ∂B/∂t and ∂D/∂t always vanish for this reason. However, the surface 596 Appendices integral of J can be nonvanishing if J diverges anywhere within the region of integration. This can occur if there is a surface current density j s , of units A/m, at the boundary between the two materials. As a consequence of these considerations, Eqs. (A.27) and (A.28) become (E 2 −E 1 ) ×n = 0, (A.31) (H 2 −H 1 ) ×n = j s . (A.32) The first of these equations states that the tangential component of E is always continuous at an interface, whereas the second states that the tangential components of H is discontinuous by an amount equal to the surface current density j s . Again, the surface current density must vanish for the interface between two dielectric media. Further reading Jackson, J.D., 1999. Classical Electrodynamics, Third Edition. Wiley, New York. Stratton, J.A., 1941. Electromagnetic Theory. McGraw–Hill, New York. Appendix B. The Gaussian System of Units In this appendix we review briefly the basic equations of electromagnetism when written in the gaussian system of units. Our treatment is a bit more abbreviated than that of Appendix A on the SI system. In the gaussian system, mechanical properties are measured in cgs units, that is, distance is measured in centimeters (cm), mass in grams (g), and time in seconds (s). The unit of force is thus the g cm/sec 2 , known as a dyne, and the unit of energy is the g cm 2 /sec 2 , known as the erg. The fundamental electrical unit is a unit of charge, known either as the statcoulomb or simply as the electrostatic unit of charge. It is defined such that the force between two charged point particles, each containing 1 statcoulomb of charge and separated by 1 centimeter, is 1 dyne. More generally, the force between two charged particles of charges q 1 and q 2 separated by the directed distance r = r ˆ r where ˆ r is a unit vector in the r direction is given by F = q 1 q 2 r 2 ˆ r. (B.1) The unit of current is thus the statcoulomb/sec, which is known as the statampere, or simply as the electrostatic unit of current. The unit of electrical poten- Appendix B. The Gaussian System of Units 597 tial (i.e., potential energy per unit charge) is the erg/statcoulomb, also known as the statvolt. In the gaussian system, Maxwell’s equations have the form ∇×E =− 1 c ∂B ∂t , (B.2a) ∇×H = 1 c ∂D ∂t + 4π c J, (B.2b) ∇·B = 0, (B.2c) ∇·D = 4πρ. (B.2d) A remarkable feature of the gaussian system is that the four primary field vectors (i.e., the electric field E, the electric displacement field D, the magnetic induction B, and the magnetic intensity H, as well as the polarization vector P and the magnetization vector M, which will be introduced shortly) all have the same dimensions—that is, [E]=[D]=[B]=[H]=[D]=[M] = statvolt cm = statcoulomb cm 2 =gauss = oersted =  erg cm 3  1/2 . (B.3) By convention the name gauss is used only in reference to the field B and oersted only with the field H. The two additional quantities appearing in Maxwell’s equations are the free charge density ρ, measured in units of statcoulomb/cm 3 , and the free current density J, measured in units of statampere/cm 2 . Under many circumstances J is given by the expression J = σ E, (B.4) which can be considered to be a microscopic form of Ohm’s law, where σ is the electrical conductivity, whose units are inverse seconds. The relationships among the four electromagnetic field vectors are known as the constitutive relations. These relations, even in the presence of nonlinearities, have the form D = E +4πP, (B.5a) H = B −4πM. (B.5b) The manner in which the response of a material medium can lead to a nonlinear dependence of P upon E is of course the subject of this book. For the limiting case of a purely linear response, the relationships can be expressed 598 Appendices (assuming an isotropic medium for notational simplicity) as P = χ (1) E, (B.6a) M = χ (1) m H. (B.6b) Note that the linear electric susceptibility χ (1) and the linear magnetic susceptibility χ (1) m are dimensionless quantities. If we now introduce the linear dielectric constant  (1) (also known as the dielectric permittivity) and the linear magnetic permeability μ (1) , both of which are dimensionless and are defined by D =  (1) E, (B.7a) B = μ (1) m H, (B.7b) we find by consistency of Eqs. (B.5a)–(B.7a) and (B.5b)–(B.7b) that  (1) = 1 +4πχ (1) , (B.8a) μ (1) = 1 +4πχ (1) m . (B.8b) The fields E and B (rather than D and H) are usually taken to constitute the fundamental electromagnetic fields. For example, the force on a particle of charge q moving at velocity v through an electromagnetic field is given by F = q  E + v c ×B  . (B.9) Poynting’s theorem can be derived from Maxwell’s equations in the following manner. We begin with the vector identity ∇·(E ×H) =H · (∇×E) −E ·(∇×H) (B.10) and introduce expressions for ∇×E and ∇×H from the Maxwell equations (B.2a) and (B.2b), to obtain c 4π ∇·(E × H) + 1 4π  H · ∂B ∂t +E · ∂D ∂t  =−J ·E. (B.11) Assuming for simplicity the case of a purely linear response, the second term on the left-hand side of this equation can be expressed as ∂u/∂t, where u = 1 8π (E ·D +B · H) (B.12) represents the energy density of the electromagnetic field. We also introduce the Poynting vector S = c 4π E ×H, (B.13) [...]... 107, 108 Impurity-doped solid 326 Index ellipsoid 513, 516, 518, 519 Instantaneous frequency 376, 377 Instantaneous response 386, 600 Intense-field nonlinear optics 561, 571, 572, 577, 586 Intensity modulator 521 Intensity-dependent refractive index 11, 12, 207, 209, 210, 213, 230, 242, 329, 369, 377 basic properties of 11, 207 Interaction picture 159, 187, 190, 192 Interfaces, nonlinear optics of 122... 473, 474 Stokes–anti-Stokes coupling 324, 488, 495, 508 Strain-optic tensor 416 Sum-frequency generation 7–9, 19, 20, 40, 41, 69, 70, 74, 78–80, 91–93, 128, 186 Supercontinuum generation 571 Surface nonlinear optics 104 Susceptibility 22, 27, 31, 32, 34, 37, 123, 135, 142, 143, 511 Index in quasi-static limit 255 linear 32, 37, 142, 143, 288, 511 calculated using density matrix 161 nonlinear 22, 31,... fields 48 Recombination, electron-hole 241, 579 Reflection, nonlinear optics in 122, 123 Refractive index, calculated quantum mechanically 223 Relativistic effects 572, 573, 580, 583, 584 relativistic change in mass 573, 581 Relaxation processes 137, 280, 281, 284, 285, 296, 327 Relaxation time 282, 392, 503, 525, 535 Residue theorem 59 Resonance, one-, two-, and three-photon 22, 136, 149, 173, 204,... picture 159 Second-harmonic generation 1, 5–8, 20, 25, 26, 39–41, 96, 97, 101–105, 120–123, 129–132 Self-action effects 209, 329–331, 333, 335, 337, 339, 341, 388, 585 Self-broadening (of atomic resonance) 283 Self-focusing 12, 329–333, 335, 337, 339–342, 383, 388, 538, 582, 583 critical power for 329, 340, 582, 585 self focusing angle 331, 332 transient 342 Self-induced transparency 387 Self-phase modulation... 600 Appendices Further reading Jackson, J.D., 1975 Classical Electrodynamics, Second Edition Wiley, New York Marion, J.B., Heald, M.A., 1980 Classical Electromagnetic Radiation Academic Press, New York Purcell, E.M., 1965 Electricity and Magnetism McGraw-Hill, New York Appendix C Systems of Units in Nonlinear Optics There are several different systems of units that are commonly used in nonlinear optics. .. Gaussian beam 250 Phase shift (as origin of two-beam coupling) 369 Phase-matching 8, 10, 69, 77, 79, 81–84, 131, 413, 414, 488 and Stokes–anti-Stokes coupling in stimulated Raman scattering 488 as the Bragg condition 416 methods of achieving 81 quasi-phase-matching 84, 85, 87, 88, 116, 129 Phonon lifetime 408, 438, 447, 460, 468 Photon energy-level 9 Photon occupation number 475–477 Photonic switching 186,... 234, 271, 392, 501 Anomalous dispersion 81 Anti-Stokes scattering 411, 412 Apparent divergence 224 Arabinose 270 Argon 283 Atomic polarizability (linear) 226 Atomic unit of electric field strength 255 vapors 21, 135, 149, 201, 283 Avalanche breakdown mechanism 544, 545, 547 B Backward light 325 Band-filling effects 244 Band-gap energy 240, 245, 369 Band-to-band transitions 241 Bands, energy 240 Barium... coefficient 41, 115 Einstein A coefficient 169, 475 Electric-dipole approximation 187 Electromagnetically induced transparency 185–187, 189, 191, 193, 203, 205 Electromagnetically induced transparency (EIT) 185 Electron-ion recombination rate 579 Electron-positron pair creation 583, 584 Electronic nonlinearities, nonresonant 221, 223, 225, 227, 228, 327 low-frequency limit 227 quantum mechanical treatment of... 168, 169, 556, 559, 579 multiphoton 543, 550, 553 N-photon 579 one-photon 553 two-photon 16, 17, 556, 559 Acceptors 526 Acetone 212, 441, 466, 469 Acoustooptics 391–428 Adiabatic following 299, 300 Air 212, 237, 395, 407, 427, 435, 544 Airy’s equation 360 Amplifier (stimulated Brillouin scattering) 430, 431, 437, 443 Analytic functions 58, 59, 62 Angle-tuned phase matching 83 Anharmonic oscillator model... 478, 493 spontaneous 17 stimulated 17, 455, 479, 493 Raman–Nath scattering (in acoustooptics) 423, 424 Raman Stokes scattering 473, 474 Raman susceptibility 483–486 Rate equation 527, 579 Rate-of-dilation tensor 457 Rayleigh resonance 325, 326 Rayleigh scattering 464, 465 spontaneous 465 stimulated 464, 465 Rayleigh-wing scattering 392, 394, 473, 501–509 polarization properties of 506–508 spontaneous . and Magnetism. McGraw-Hill, New York. Appendix C. Systems of Units in Nonlinear Optics There are several different systems of units that are commonly used in nonlinear optics. In this appendix. any well-behaved vector field A, the following identity holds:  V ∇·A dV =  S A ·n da. (A.23) 594 Appendices The integral on the left-hand side is to be performed over a closed three- dimensional. 169, 556, 559, 579 multiphoton 543, 550, 553 N-photon 579 one-photon 553 two-photon 16, 17, 556, 559 Acceptors 526 Acetone 212, 441, 466, 469 Acoustooptics 391–428 Adiabatic following 299, 300 Air