Multipole radiation
Multipole radiation is a theoretical framework for the description of electromagnetic or gravitational radiation from time-dependent distributions of distant sources. These tools are applied to physical phenomena which occur at a variety of length scales - from gravitational waves due to galaxy collisions to gamma radiation resulting from nuclear decay. Multipole radiation is analyzed using similar multipole expansion techniques that describe fields from static sources, however there are important differences in the details of the analysis because multipole radiation fields behave quite differently from static fields. This article is primarily concerned with electromagnetic multipole radiation, although the treatment  of gravitational waves is similar. Electromagnetic radiation depends on structural details of the source system of electric charge and electric current. Direct analysis can be intractable if the structure is unknown or complicated. Multipole analysis offers a way to separate the radiation into moments of increasing complexity. Since the electromagnetic field depends more heavily on lower-order moments than on higher-order moments, the electromagnetic field can be approximated without knowing the structure in detail. Properties of multipole radiation. Linearity of moments. Since Maxwell's equations are linear, the electric field and magnetic field depend linearly on source distributions. Linearity  allows the fields from various multipole moments to be calculated independently and added together to give the total field of the system. This is the well-known principle of superposition. Origin dependence of multipole moments. Multipole moments are calculated with respect to a fixed expansion point which is taken to be the origin of a given coordinate system. Translating the origin changes the multipole moments of the system with the exception of the first non-vanishing moment. For example, the monopole moment of charge is simply the total charge in the system. Changing the origin will never change this moment . If the monopole moment is zero then the dipole moment of the system will be translation invariant. If both the monopole and dipole moments are zero then the quadrupole moment is translation invariant, and so forth. Because higher-order moments depend on the position of the origin, they cannot be regarded as invariant properties of the system. Field dependence on distance. The field from a multipole moment depends on both the distance from the origin and the angular orientation of the evaluation point with respect to the coordinate system. In particular, the radial dependence of the electromagnetic field from  a "stationary" formula_1-pole scales as formula_2. That is, the electric field from the electric monopole moment scales as inverse distance squared. Likewise, the electric dipole moment creates a field that scales as inverse distance cubed, and so on. As distance increases, the contribution of high-order moments becomes much smaller than the contribution from low-order moments, so high-order moments can be ignored to simplify calculations. The radial dependence of radiation waves is different from static fields because these waves carry energy away from the system. Since energy must be conserved, simple geometric analysis shows that the energy  density of spherical radiation, radius formula_3, must scale as formula_4. As a spherical wave expands, the fixed energy of the wave must spread out over an expanding sphere of surface area formula_5. Accordingly, every time-dependent multipole moment must contribute radiant energy density that scales as formula_4, regardless of the order of the moment. Hence, high-order moments cannot be discarded as easily as in static case. Even so, the multipole coefficients of a system generally diminish with increasing order, usually as formula_7, so radiation fields can still be approximated by truncating high-order moments. Time-dependent electromagnetic fields . Sources. Time-dependent source distributions can be expressed using Fourier analysis. This allows separate frequencies to be analyzed independently. Charge density is given by formula_8 and current density by formula_9 For convenience, only a single angular frequency ω is considered from this point forward; thus formula_10 formula_11 The superposition principle may be applied to generalize results for multiple frequencies. Vector quantities appear in bold. The standard convention of taking the real part of complex quantities to represent physical quantities is used. The intrinsic angular momentum of elementary particles (see Spin (physics)) may also affect electromagnetic radiation from some source materials . To account for these effects, the intrinsic magnetization of the system formula_12 would have to be taken into account. For simplicity however, these effects will be deferred to the discussion of generalized multipole radiation. Potentials. The source distributions can be integrated to yield the time-dependent electric potential and magnetic potential φ and A respectively. Formulas are expressed in the Lorenz Gauge in SI units. formula_13 formula_14 In these formulas "c" is the speed of light in vacuum, formula_15 is the Dirac delta function, and formula_16 is the Euclidean distance from the source point x′ to the evaluation point x.  Integrating the time-dependent source distributions above yields formula_17 formula_18 where . These formulas provide the basis for analyzing multipole radiation. Multipole expansion in near field. The near field is the region around a source where the electromagnetic field can be evaluated quasi-statically. If target distance from the multipole origin formula_19 is much smaller than the radiation wavelength formula_20, then formula_21. As a result, the exponential can be approximated in this region as: formula_22 See Taylor expansion. By using this approximation, the remaining dependence is the same as it is for a static system, the same analysis applies. Essentially, the  potentials can be evaluated in the near field at a given instant by simply taking a snapshot of the system and treating it as though it were static - hence it is called quasi-static. See near and far field and multipole expansion. In particular, the inverse distance formula_23 is expanded using spherical harmonics which are integrated separately to obtain spherical multipole coefficients. Multipole expansion in far field: Multipole radiation. At large distances from a high frequency source, formula_24, the following approximations hold: formula_25 formula_26 Since only the first-order term in formula_27 is significant at large distances, the expansions combine  to give formula_28 Each power of formula_29 corresponds to a different multipole moment. The first few moments are evaluated directly below. Electric monopole radiation, nonexistence. The zeroth order term, formula_30, applied to the scalar potential gives formula_31 where the total charge formula_32 is the electric monopole moment oscillating at frequency . Conservation of charge requires since formula_33 If the system is closed then the total charge cannot fluctuate which means the oscillation amplitude "q" must be zero. Hence, formula_34. The corresponding fields and radiant power must also be zero. Electric dipole radiation. Electric dipole potential. Electric dipole radiation can be  derived by applying the zeroth-order term to the vector potential. formula_35 Integration by parts yields formula_36 and the charge continuity equation shows formula_37 It follows that formula_38 Similar results can be obtained by applying the first-order term, formula_39 to the scalar potential. The amplitude of the electric dipole moment of the system is formula_40, which allows the potentials to be expressed as formula_41 formula_42 Electric dipole fields. Once the time-dependent potentials are understood, the time-dependent electric field and magnetic field can be calculated in the usual way. Namely, formula_43 formula_44 or, in a source-free region  of space, the relationship between the magnetic field and the electric field can be used to obtain formula_45 formula_46 where formula_47 is the impedance of free space. The electric and magnetic fields that correspond to the potentials above are formula_48 formula_49 which is consistent with spherical radiation waves. Pure electric dipole power. The power density, energy per unit area per unit time, is expressed by the Poynting vector formula_50. It follows that the time averaged power density per unit solid angle is given by formula_51 The dot product with formula_52 extracts the emission magnitude and the factor of 1/2 comes  from averaging over time. As explained above, the formula_53 cancels the radial dependence of radiation energy density. Application to a pure electric dipole gives formula_54 where θ is measured with respect to formula_55. Integration over a sphere yields the total power radiated: formula_56 Magnetic dipole radiation. Magnetic dipole potential. The first-order term, formula_39, applied to the vector potential gives magnetic dipole radiation and electric quadrupole radiation. formula_58 The integrand can be separated into symmetric and anti-symmetric parts in J and x′ formula_59 The second term contains the effective magnetization due to the current formula_60 and integration gives the  magnetic dipole moment. formula_61 formula_62 Notice that formula_63 has a similar form to formula_64. That means the magnetic field from a magnetic dipole behaves similarly to the electric field from an electric dipole. Likewise, the electric field from a magnetic dipole behaves like the magnetic field from an electric dipole. Taking the transformations formula_65 formula_66 formula_67 on previous results yields magnetic dipole results. Magnetic dipole fields. formula_68 formula_69 Pure magnetic dipole power. The average power radiated per unit solid angle by a magnetic dipole is formula_70 where θ is measured with respect to the magnetic dipole formula_71. The total power  radiated is: formula_72 Electric quadrupole radiation. Electric quadrupole potential. The symmetric portion of the integrand from the previous section can be resolved by applying integration by parts and the charge continuity equation as was done for electric dipole radiation. formula_73 formula_74 This corresponds to the traceless electric quadrupole moment tensor formula_75. Contracting the second index with the normal vector formula_76 allows the vector potential to be expressed as formula_77 Electric quadrupole fields. The resulting magnetic and electric fields are: formula_78 formula_79 Pure electric quadrupole power. The average power radiated per unit solid angle by an electric quadrupole is formula_80 where  θ is measured with respect to the magnetic dipole formula_71. The total power radiated is: formula_82 Generalized multipole radiation. As the multipole moment of a source distribution increases, the direct calculations employed so far become too cumbersome to continue. Analysis of higher moments requires more general theoretical machinery. Just as before, a single source frequency formula_83 is considered. Hence the charge, current, and intrinsic magnetization densities are given by formula_84 formula_85 formula_86 respectively. The resulting electric and magnetic fields share the same time-dependence as the sources. formula_87 formula_88 Using these definitions and the continuity equation allows Maxwell's equations  to be written as formula_89 formula_90 formula_91 formula_92 These equations can be combined by taking the curl of the last equations and applying the identity formula_93. This gives the vector forms of the non-homogeneous Helmholtz equation. formula_94 formula_95 Solutions of the wave equation. The homogeneous wave equations that describes electromagnetic radiation with frequency formula_83 in a source-free region have the form. formula_97 The wave function formula_98 can be expressed as a sum of vector spherical harmonics formula_99 formula_100 Where formula_101 are the normalized vector spherical harmonics and formula_102 and formula_103 are spherical Hankel functions. See spherical Bessel functions . The differential operator formula_104 is the angular momentum operator with the property formula_105. The coefficients formula_106 and formula_107 correspond to expanding and contracting waves respectively. So formula_108 for radiation. To determine the other coefficients, the Green's function for the wave equation is applied. If the source equation is formula_109 then the solution is: formula_110 The Green function can be expressed in vector spherical harmonics. formula_111 Note that formula_112 is a differential operator that acts on the source function formula_113. Thus, the solution to the wave equation is: formula_114 Electric multipole fields. Applying the above solution to the electric multipole  wave equation formula_115 gives the solution for the magnetic field: formula_116 formula_117 The electric field is: formula_118 The formula can be simplified by applying the identities formula_119 formula_120 formula_121 to the integrand, which results in formula_122 Green's theorem and integration by parts manipulates the formula into formula_123 The spherical bessel function formula_124 can also be simplified by assuming that the radiation length scale is much larger than the source length scale, which is true for most antennas. formula_125 Retaining only the lowest order terms results in the simplified form for the electric multipole coefficients: formula_126 formula_127 formula_128 formula_129 is  the same as the electric multipole moment in the static case if it were applied to the static charge distribution formula_130 whereas formula_131 corresponds to an induced electric multipole moment from the intrinsic magnetization of the source material. Magnetic multipole fields. Applying the above solution to the magnetic multipole wave equation formula_132 gives the solution for the electric field: formula_133 formula_134 The magnetic field is: formula_135 As before, the forumula simplifies to: formula_136 Retaining only the lowest order terms results in the simplified form for the magnetic multipole coefficients: formula_137 formula_138 formula_139 formula_140 is the magnetic multipole moment from the  effective magnetization formula_141 while formula_142 corresponds to the intrinsic magnetization formula_143. General solution. The electric and magnetic multipole fields combine to give the total fields: formula_144 formula_145 Note that the radial function formula_146 can be simplified in the far field limit formula_147. formula_148 Thus the radial dependence of radiation is recovered.