Angular Momentum and Conservation Laws in Classical Electrodynamics

Transcription

1 UPTEC F Examensarbete 30 hp Februari 2009 Angular Momentum and Conservation Laws in Classical Electrodynamics Johan Lindberg

2 Abstract Angular Momentum and Conservation Laws in Classical Electrodynamics Johan Lindberg Teknisk- naturvetenskaplig fakultet UTH-enheten Besöksadress: Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0 Postadress: Box Uppsala Telefon: Telefax: Hemsida: In analogy with mechanics the concepts of field momentum and field angular momentum can be introduced in electrodynamics through conservation laws. In this thesis some fundamental properties of the electromagnetic field angular momentum are treated. This includes the connection to sources, conservation laws and the separation of total angular momentum into a spin and an orbital part. In 1957 Zeldovich pointed out that particles with spin 1/2 must possess, in addition to a magnetic moment, another dipole characteristic. He called it the anapole. Zeldovich's idea was theoretically clarified and generalized in the 1970's and an entire class of moments were introduced. These were called the toroidal multipoles and they constitute a family of multipoles independent of electric and magnetic moments. Their relation to the electromagnetic angular momentum is discussed, however, as the standard procedure found in most textbooks implicitly contains these multipoles, their introduction do not seem to yield fundamentally new results. The fields, intensity and angular momentum from a toroidal dipole is given in this thesis. A new expression for electromagnetic angular momentum in terms of retarded integrals of the sources is given, and the angular momentum density from an electric dipole is calculated using this expression. Research on the photon orbital angular momentum is a hot topic today with applications in various fields. Recent publications suggest the possibility to use the orbital angular momentum of the electromagnetic fields in radio. Conservation laws for fields in an orbital angular momentum eigenstate are derived, where analogies with the linear momentum and stress tensor, in terms of the eigenmodes, are introduced. A separation of angular momentum into its spin and orbital parts would have a major impact on the applications of orbital angular momentum since then the orbital structure of the field could be measured directly. A complex Lagrangian density is constructed from the complex field strength tensor and its dual tensor. The corresponding field equations are Maxwell's equations in Majorana form, expressed in Riemann-Silberstein vectors. The conservation laws due to translational, Lorentz, and phase transformations are derived using Noether's theorem. The symmetry of translational invariance yields the Maxwell stress tensor. However, within the expression provided by Noether's theorem, another rank two tensor, which describes the (wave polarization of the electromagnetic fields, also appears but ultimately cancels. The conservation law can be interpreted as the sum of two separate conservation laws, one for right-handed and one for left-handed fields, which appear due to the same symmetry. However, as this conservation law appears due to translational rather than rotational invariance, the separation of angular momentum remains a mystery. Handledare: Bo Thidé Ämnesgranskare: Bo Thidé Examinator: Tomas Nyberg ISSN: , UPTEC F09 017

3 DIPLOMA THESIS ANGULAR MOMENTUM AND CONSERVATION LAWS IN CLASSICAL ELECTRODYNAMICS JOHAN LINDBERG Uppsala School of Engineering and Department of Astronomy and Space Physics, Uppsala University, Sweden FEBRUARY 16, 2009

4 since that time, sun, moon, and stars may pursue their course I know not whether it is day or night the whole world is nothing to me The Sorrows of Young Werther, Goethe

5 CONTENTS Contents Preface Acknowledgments iii v vii 1 Classical Electrodynamics Conservation Laws in Classical Electrodynamics Complex Fourier Fields 3 2 Multipole Expansion Transverse Fields Expansion of the Sources Introduction of Mean Radii Expansion of the Current Introduction of Toroidal Moments Radiation Multipole Radiated Angular Momentum Fields and Angular Momentum from a Toroidal Dipole 15 3 General Expression for Angular Momentum Arbitrary Source Distributions Angular Momentum from an Electric Dipole 22 4 Conservation Laws for Orbital Angular Momentum The Corresponding Maxwell Equations Pure Eigenstates Poynting s Theorem for Pure OAM Modes The Energy-Momentum Theorem for Pure OAM Modes Conservation Law of Angular Momentum for Pure OAM Modes 37 5 Covariant Lagrangian RS Electrodynamics 39 iii

6 CONTENTS Relation to Zilch Covariant Riemann-Silberstein Formulation Construction of a New Lagrangian Noether s Theorem Conserved Quantities Lorentz Transformations Additional Symmetry 50 6 Discussion Multipoles The General Expression for Angular Momentum Conservation Laws for Orbital Angular Momentum Lagrangian Riemann-Silberstein Electrodynamics 54 Bibliography 57 iv

7 PREFACE The subject of electromagnetic angular momentum is not a trivial one. The concept of orbital angular momentum (OAM is not an exception, and is in general merely mentioned in most elementary texts on electrodynamics. The electric fields can be thought of as quantum wave functions for photons [10]. In these terms the angular momentum consists of a spin and an orbital part. However, these components differ in an important and fundamental way. Whereas the spin angular momentum has two orthogonal states, the (photon orbital angular momentum consists of a denumerably infinite number of discrete orthogonal states. Poynting suggested that circularly polarized light should carry spin angular momentum [17] and Allen et al. [1] proposed that light beams with a helical phase dependence carry orbital angular momentum, a physical observable that is independent of the spin angular momentum. This has become an important research field with applications in astronomy and astrophysics [13, 10], but the areas studied range from wireless communication concepts [12] to quantum entanglement and quantum computation [29]. The possibility to utilize the orbital angular momentum in radio science is discussed in Ref. [28]. E.g., the orbital angular momentum could play an important role for future radio communications. This thesis is an investigation of the electromagnetic field angular momentum regarding its relation to sources, conservation laws and the separation into spin and orbital angular momentum. Structure The first part, Chapters 1 2, consists of an introduction to classical electrodynamics, followed by a discussion of the multipole expansion by Dubovik et al. [8, 9] which introduces the toroidal multipole moments, and their impact on the angular momentum. The total fields, radiation, intensity and angular momentum from a toroidal dipole is calculated. Whereas expressions for the angular momentum from multipoles of an arbitrary order can be found [19], in what way is it possible to relate this property of the fields to the source distribution? A general expression for the electromagnetic angular momentum in terms of retarded integrals of the sources can be found in Chapter 3, and the angular momentum from an electric dipole is calculated using this expression. The article by Bergman et al. [4] shows that the (wave polarization of electromagnetic fields fulfils a conservation law of its own. The question is then whether the orbital v

8 CONTENTS angular momentum states obey some conservation laws as well. In Chapter 4 the analogies to Poynting s theorem, the energy-momentum theorem and the conservation law for angular momentum are derived for orbital angular momentum eigenstates. The classical spin of electromagnetic fields, i.e. their polarization, is the expectation value of the quantum mechanical vector operator for spin angular momentum [22]. The possibility to separate the total angular momentum would mean that the orbital part of electromagnetic fields can be measured directly. By adopting a similar approach as in Ref. [4], and the formulation of electrodynamics in terms of the Riemann-Silberstein vectors, a complex Lagrangian density is constructed. From this we attempt to derive the conservation law of polarization covariantly in Chapter 5. An investigation of the symmetry properties from which this law originates could possibly answer the question of how to relate the polarization to the total angular momentum. vi

9 ACKNOWLEDGMENTS I would very much like to thank my supervisor, professor Bo Thidé, as well as Dr. Holger Then, for their guidance and for leading me into the unexplored lands of modern research. I am most thankful to Dr. Jan Bergman for many interesting discussions and valuable comments. A very special thanks to Siavoush Mohammadi for sharing his knowledge, about so much beyond electrodynamics, but mostly for his ever so kindly friendship and tireless assistance this autumn. For all the help I got without ever asking for it. Thank you. To Maja Llena Garde for spreading joy around the office. Always in such a good mood, always there to talk to. Of course, there are people left to thank, the staff at the Swedish Institute of Space physics and the Department of Physics and Astronomy at Uppsala, my fellow thesis students etc.. Lastly, to the people in my life that are close to me, I am always grateful for you ever being there when needed. vii

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11 1 CLASSICAL ELECTRODYNAMICS Using an axiomatic approach, one can consider Classical Electrodynamics (CED to be governed by Maxwell s equations (ME, d 2 x ˆn E = d 3 x ρ, (1.1 ɛ 0 S S dl E = V S d 2 x ˆn B t, (1.2 d 2 x ˆn B = 0, (1.3 ( dl B = d 2 x ˆn µ 0 j + 1 E S c 2, t (1.4 which are the dynamical relations between the two vector fields E and B, their sources ρ and j and the two scalars µ 0 and ɛ 0, called the electric field, the magnetic field, the electric charge density, the electric current, the permittivity of free space, and the permeability of free space, respectively. Using the divergence theorem and Stokes theorem, the Maxwell equations can be written in differential form as E = ρ ɛ 0, (1.5 E = B t, (1.6 B = 0, (1.7 B = µ 0 j + 1 c 2 E t. (1.8 The Maxwell equations are the fundamental equations describing all classical electromagnetic phenomena. From ME the following two important electrodynamical relations 1

12 CHAPTER 1. CLASSICAL ELECTRODYNAMICS can be derived, namely the continuity equation ρ + j = 0, (1.9 t which describes the conservation of electric charge, and the Lorentz force F M = qe + j B. (1.10 The corresponding Lorentz force density will be written as t p M = f M = ρe + j B (1.11 which is the time rate of change of the matter momentum density p M. 1.1 Conservation Laws in Classical Electrodynamics Two important conservation laws in electrodynamics are Poynting s theorem and the energy-momentum theorem. In differential form, Poynting s theorem is given by [14] u + S + w = 0, (1.12 t where u = ɛ 0 ( E 2 + c 2 B 2 /2 is the electromagnetic energy density, and w = j E is the rate of work density done on matter by the fields and the vector S = (E B/µ 0 is called the Poynting vector. This is a conservation law describing the flow of electromagnetic energy, with the Poynting vector describing the energy flux. These interpretations can be concluded by considering the integral form of this conservation law d d 3 xu + d 2 x ˆn S + d 3 xw = 0, (1.13 dt V S V stating that the rate of change of energy within the volume is given by the flux of energy across the boundary and the rate of work done within the same volume. The energy-momentum theorem reads [14] where p M t + 1 c 2 S t + T M = 0, (1.14 T M = ɛ 0 [ 1 2 ] ( E 2 + c 2 B 2 I E E c 2 B B (1.15 2

13 1.2. COMPLEX FOURIER FIELDS denotes the Maxwell stress tensor and I is the 3D unit tensor. In integral form Eq. (1.14 reads d dt Vd 3 xp M + d d 3 xp field + d 2 x ˆn T M = 0 (1.16 dt V S where the electromagnetic field momentum p field = S/c 2 has been introduced. This conservation law then states that the change in total momentum within a volume V is given by the flow of momentum across the boundary. Less well known is the angular momentum conservation law, given differentially by r t p M + r t p field + (r T M = 0, (1.17 which in integral form reads d dt Vd 3 xr p M + d d 3 xr p field + d 2 x ˆn (r T M = 0. (1.18 dt V S This can be interpreted such that the change in total angular momentum within a volume V is given by the flow of angular momentum across the volume boundary. With this interpretation the concept of electromagnetic angular momentum has been introduced by h = r p field = ɛ 0 r (E B. ( Complex Fourier Fields The electric and magnetic fields are generally both space and time dependent, i.e. E = E(t,r and B = B(t,r. However, as time is singled out as a compared to the three spatial components, it is convenient to express the fields in terms of their temporal Fourier transforms F[E(t,r] E ω (r = 1 2π F[B(t,r] B ω (r = 1 2π dt E(t,re iωt, (1.20 dt B(t,re iωt, (1.21 and consider E ω and B ω as complex vectors. The introduction of complex fields is just a mathematical tool and usually only the real part of the complex quantities have any physical meaning. This technique is often very helpful and in a way seems to be a more natural way to describe electrodynamics since it simplifies the mathematics. However, 3

14 CHAPTER 1. CLASSICAL ELECTRODYNAMICS care must be exercised when multiplying two such complex quantities since only the real part of their expressions are intended as having physical meaning, hence J(t,r E(t,r = Re{J(t,r} Re{E(t,r} = 1 [ Jω (re iωt + J 4 ω(re iωt] [E ω (re iωt + E ω(re iωt] = 1 2 Re{ J ω(r E ω (r + J ω (r E ω (re 2iωt} = 1 2 Re{ J(t,r E(t,r + J(t,r E(t,r }, (1.22 where * denotes complex conjugate. However, when averaging over time, the second term of Eq. (1.22 vanishes. In other words, for the scalar product of two complex fields only the following relation is frequently used [14] J(t,r E(t,r = 1 2 Re{ J ω(r E ω (r } R, (1.23 with time averaging understood. This is often the case when a harmonic time dependence of the fields is assumed, which means that E(t,r = E ω (re iωt = E ω e iωt. This also corresponds to fields having only a single temporal Fourier component. Poynting s Theorem for Complex Fields By writing the fields as complex Fourier components, E(t,r = E ω (re iωt, (1.24 B(t,r = B ω (re iωt, (1.25 the differential, time averaged Poynting s theorem (in vacuum takes the form 1 2 Re{ j } 1 ω E ω + Re { E ω B } ω = 0. (1.26 2µ 0 The real valued vector Re { E ω B ω} /2µ0 is often referred to as the Poynting vector as well. 4

15 2 MULTIPOLE EXPANSION A multipole expansion is the expansion of a physical quantity into a general infinite series in a complete mathematical basis. The reason for using a multipole expansion in CED is to approximate the contribution from complicated distributions of currents and charges by (a series of quantities that are easier to calculate. Indeed, as higher and higher terms (with respect to the expansion parameter in the expansion are considered, the terms become more complicated and difficult to deal with, but they usually contribute gradually less to the total value of the series. A multipole expansion of time dependent sources is performed in most textbooks on electrodynamics (e.g. Ref. [14] and has been a cornerstone of the electrodynamical theory since its foundations were laid in the 19th century. Yet, the fairly recent results of Dubovik et al. [8, 9] show that the multipole expansions, outlined in most textbooks, have flaws. Probably the most serious one is that they neglect the existence of toroidal moments. This is a whole family of multipoles, completely independent of conventional electric and magnetic moments. The multipole expansion is no different from any expansion in a complete mathematical basis. The coefficients before each basis function are called the multipoles. Their explicit appearances therefore depend on the specific basis chosen. However, in the general case, a spherical basis is most often used. One important reason for this is that the expansion coefficients in this basis become irreducible. It is, of course, possible to find the irreducible moments as combinations of the expansion coefficients in any basis. However, by choosing a spherical basis, we retrieve them immediately. The violation of parity was established in weak interaction theory in the 1950 s. Zeldovich [2] suggested that a third form factor in the parametrization of the Dirac spinor particle current should be introduced. He called the classical counterpart to this form factor the anapole. Ever since then Russian authors have been investigating the multipole expansion, in both classical and quantum field theory [8, 9, 19, 20]. In 1974 Dubovik generalized Zeldovich s idea and introduced the toroidal multipole moments and form factors. 5

16 CHAPTER 2. MULTIPOLE EXPANSION The main focus regarding the multipole expansion introduced by Dubovik et al. [8, 9] is on the introduction of the toroidal moments and what impact of their introduction has on the theory of electromagnetic angular momentum. 2.1 Transverse Fields In the time dependent case, when radiation is taken into account, the notation of transverse electric (TE and transverse magnetic (TM fields can be introduced. This is due to the fact that in the sourcefree regions Maxwell s equations take the form E = 0, B = 0, E = B t, (2.1 B = 1 c 2 E t. (2.2 If a harmonic time dependence is assumed, i.e. E(t,r = E ω (re iωt and B(t,r = B ω (re iωt, then ME become E ω = 0, E ω = iωb ω = ikcb ω, (2.3 B ω = 0, B ω = i k c E ω. (2.4 From these equations the electric or the magnetic fields can be eliminated to yield the two sets of equations ( 2 + k 2 B ω = 0, B ω = 0, E ω = ic k B ω, In order to proceed, the vector relation 2 (r V = r ( 2 V + 2 V, and ( 2 + k 2 E ω = 0, E ω = 0, B ω = i ck E ω. can be used to rewrite the homogeneous Helmholtz equation (HHE of Eq. (2.5 as { ( 2 + k 2 (r B ω = 0, ( 2 + k 2 (r E ω = 0. (2.5 (2.6 Being a solution of the HHE, the scalar functions r B ω and r E ω can now be expressed in the spherical basis. We thus define a transverse electric (TE multipole field of order (l,m by the conditions [14] r B lm ω (M = r E lm ω (M = 0. l(l + 1 µ 0 k g l(kry lm (θ,ϕ, 6

17 2.2. EXPANSION OF THE SOURCES Normally g l = A (1 l h (1 l (kr+a (2 l h (2 l (kr, depending on the boundary conditions. However, only outgoing waves will be considered. Thus the solutions proportional to h (1 l, which are the spherical Hankel functions defined as h l (a = π a 1 2 H l+ 1 (a [14], will be used. 2 We can now relate r B ω to the electric field by ckr B ω = 1 i r ( E ω = 1 i (r E ω = L E ω, (2.7 where the orbital angular momentum operator L = ir has been introduced. Hence, L E lm ω (M = cl(l + 1h (1 l (kry lm (θ,ϕ. Then for a TE E lm ω (M = ch (1 l (krly lm (θ,ϕ, (2.8 B lm ω (M = i ck Elm ω (M. (2.9 Reasoning as above, we define a transverse magnetic (TM multipole field by E lm ω (E = ic k Blm ω (E, B lm ω (E = h (1 l (krly lm (θ,ϕ. Introducing the vector spherical harmonic [5] Y l,l,m = 1 l(l+1 LY lm (θ,ϕ, a general solution of ME can be written as a superposition of TE and TM fields B ω = µ 0 l,m a E (l,mh (1 l (kry l,l,m i [ ] k a M(l,m h (1 l (kry l,l,m, (2.10 µ0 i [ ] E ω = ɛ 0 l,m k a E(l,m h (1 l (kry l,l,m + a M (l,mh (1 l (kry l,l,m. (2.11 This multipole expansion of time dependent fields can be found in Ref. [14]. There is nothing incorrect in this expansion; in fact this expansion contains the toroidal sources as well. It is the identification of the coefficients a E (l,m in the limit k 0 with only the charge multipoles that is incorrect. 2.2 Expansion of the Sources Dubovik s [8] idea was to explicitly expand the sources in the form of a sum of point sources. Mathematically this means an expansion in a complete set of δ-like functions, 7

18 CHAPTER 2. MULTIPOLE EXPANSION where δ is the Dirac delta function. The sources therefore become distributions and only have a meaning when used in an integral. The idea can be realized in the following way. Consider an arbitrary system described by a charge density ρ(t,r. By writing ρ in the form ρ(t,r = x ρ(t,rδ(r r, (2.12 V d3 the δ function can be expanded in a Taylor series δ(r r = l=0 ( 1 l x i x l! k i k δ(r, with summation over repeated indices. The charge density can then be written as ρ(t,r = l=0 ( 1 l l! A (l i k i k δ(r, where the tensors A (l i k = d 3 xρ(t,rx i x k V are the corresponding multipole moments. However, instead we will use the fact that the spherical basis satisfies a completeness relation. The spherical basis is the regular (that is, finite at the origin solution to the homogeneous Helmholtz equation. This is the basis given by [14] F lmk (r = f l (kry lm (ˆr, (2.13 where Y lm (θ,ϕ and f l are the spherical harmonics and the spherical Bessel functions respectively. The spherical Bessel functions are defined as in Ref. [8] and Ref. [19] by f l (x = (2π 3 2 i l π 2 I l x, (2.14 where I l+ 1 are the usual Bessel functions of half-integer order. Note that this definition 2 coincides with that in Ref. [14] except for a normalization factor of 4πi l. With the normalization of Eq. (2.14 the F lmk functions are orthogonal in the sense that [8] V d 3 x F lmk (rfl (2π3 m k (r = k 2 δ l lδ m mδ(k k, and satisfy the completeness relation [8] F lmk (rflmk(r = (2π 3 δ(r r, (2.15 k,l,m 8

19 2.2. EXPANSION OF THE SOURCES where k = 0 k2 dk. Substituting this in Eq. (2.12 yields where ρ(t,r = x ρ(t,r δ(r r = 1 V d3 (2π 3 q lm F lmk (r, (2.16 k,l,m q lm = d 3 x Flmk(rρ(t,r. (2.17 V For small values of kr the spherical Bessel function can be approximated by [14] F lmk (r = f l (kry lm (θ,ϕ = (1 4π(ikrl (kr2 (2l + 1!! 2(2l Y lm (θ,ϕ 4π(ikrl (2l + 1!! Y lm(θ,ϕ, (2.18 where!! is the semi-factorial operator, satisfying (2n + 1!! = (2n + 1!/(2 n n!. When comparing the definition of the electrostatic multipole charge moments with Q lm = 4π d 3 xy 2l + 1 lm(θ,ϕr l ρ(r, V q lm = d 3 x 4π( ikrl V (2l + 1!! Y lmρ(t,r, (2.19 we see that in the limit k 0 the quantity (the notation is due to the convention in, e.g., Refs. [8], [9], [19], [20] Q lm ( k 2,t = (2l + 1!! ( ik l 4π(2l + 1 q lm (2.20 coincide with the usual, time dependent, charge multipole moment. That is, Q lm (0,t = Q lm (t. We call these quantities the charge multipole form factors. The charge density can now be expressed as ρ(t,r = 1 4π(2l + 1 (2π 3 ( ik l F lmk (rq lm ( k 2,t. (2.21 (2l + 1!! l,m,k 9

20 CHAPTER 2. MULTIPOLE EXPANSION Introduction of Mean Radii The general dependence on k is treated by expanding the form factors Q lm ( k 2,t in their Taylor expansion with respect to ( k 2, Q lm ( k 2,t = Q (n lm = n=0 ( k 2 n Q (n lm n! = Q ( k lm(t + 2 n Q (n lm n=1 n!, (2.22 d n d( k 2 n Q lm. (2.23 k 2 =0 The terms of the charge density involving the derivatives of the form factors are called the 2n-mean radii of the 2 l -pole charge distribution. To find their explicit expressions, we insert the expansion of Q lm into Eq. (2.23 and multiply by a suitable normalizing factor [19], rlm 2n(t = 2n (2l + 2n + 1!! Q (n 4π lm (2l + 1!! = d 3 xr l+2n Y 2l + 1 lmρ(t,r. (2.24 V The normalization constant arises from the derivative of the spherical Bessel function with respect to ( k 2. These quantities describe the spatial extent of the corresponding multipole distribution [9]. For example, the simplest radius is that of the total charge r (2 00 = d 3 xρ(t,rr 2, (2.25 V so that the mean-square radius of a dipole distribution is r (2 d = V d3 xr 2 rρ(t,r and the 2n-radius of a dipole distribution is simply r (2n d = d 3 x r 2n rρ(t,r. (2.26 V Expansion of the Current Similarly, the current can be expanded in a basis of orthogonal functions. To achieve this, consider first the Helmholtz theorem, which states that any general real, well-behaved vector field can be decomposed into a transverse and a longitudinal part: j = j + j, (2.27 which is usually written j = Φ + A. With transverse and longitudinal we mean j = 0, (2.28 j = 0. (

21 2.2. EXPANSION OF THE SOURCES However, the decomposition can be pursued even further [9, 23]. The transverse part can be written as the sum of a poloidal and a toroidal part in the following way: j = r ψ (r χ = (rψ + (rχ, (2.30 where χ and ψ are scalar functions. This was used in TE and TM decomposition of the electric and magnetic fields earlier, where the transverse part vanished since source free regions were considered. It is the first term in Eq. (2.30 that describes toroidal currents (currents flowing along the axis of a torus and the second field describes poloidal currents (currents flowing around the surface of the torus. Now, just as the charge density was expanded, the three scalar functions Φ, ψ and χ can be expanded in the same way. The multipole expansion of the current and charge distributions, and hence the electromagnetic fields, has therefore been reduced to the expansion of scalar functions. Just as for the charge moments, the current is expanded in coefficients depending on k. By investigating how the expansion coefficients behave in the limit of point sources, i.e. in the limit k 0, we can relate them to the common multipole moments. From the above it can be seen that the decomposition of the current requires the introduction of three scalar functions, which, when expanded in the spherical basis, introduce three sets of expansion coefficients. The continuity equation, ρ t + j = 0 (2.31 puts a restriction only on the longitudinal part of the current, which means that the expansion coefficients, the multipole form factors, can be denoted Q lm ( k 2,t, M lm ( k 2,t and E lm ( k 2,t. The form factors M lm ( k 2,t are called the magnetic form factors and reduce in the limit k 0 to the time dependent magnetic multipoles [8] M lm ( k 2 i(2l + 1!!,t = ( ik l d 3 x fl Y l,l,m j, (2.32 4π(2l + 1(l + 1/l V i 4πl M l,m (0,t = d 3 xr l Y (2l + 1(l + 1 l,l,m j. (2.33 V Just as the charge form factors can be expanded in ( k 2, we can expand M lm ( k 2,t in the same way. This introduces the magnetic mean radii, whose explicit expressions can be found in Ref. [19]. The E lm coefficients, called the electric form factors, and the M lm coefficients are the analogues to the expansion of the fields in the a E and a M coefficients Introduction of Toroidal Moments The original work by Dubovik et al. [8, 9] was to explain how the electric form factors are related to the charge form factors. In Ref. [8] it is used that the E lm form factors reduce 11

22 CHAPTER 2. MULTIPOLE EXPANSION to Q lm (0,t of order k l 1, while the remaining part goes as k l+1, which is independent of Q lm (0,t [18]. This independent part is defined as T lm ( k 2,t = E lm( k 2,t Q lm (0,t k 2, (2.34 where the quantities T lm ( k 2,t are called the toroidal multipole form factors. They are given explicitly by [8] T lm ( k 2,t = [ (2l + 1!! ] ly 4π(l + 1/l ( ik l+1 d 3 l + 1 x l,l+1,m fl+1 ikr fl Y l,l 1,m j. (2l + 1 V 2 (2.35 In analogy, the toroidal multipole moments are defined as the form factors in the limit k 0, [ 4πl T lm (t = d 3 xr l+1 Y l,l 1,m + 2 l/(l + 1 2(2l + 1 2l + 3 V Y l,l+1,m Expanding the toroidal form factors with respect to ( k 2, we obtain T lm ( k 2,t = n ( k 2 n n! T (n where T (n lm (t = d n T d( k 2 n lm ( k 2,t in Ref. [9]. lm (t = T lm(0,t + k 2 =0 n=1 ( k 2 n n! ] j (2.36 T (n lm (t, (2.37. The expression for their mean radii can be found 2.3 Radiation The expressions for the electric and the magnetic fields in terms of multipoles and form factors can be found from the potentials. They can in turn be found from the retarded integrals φ ω = 1 V d3 x ρ(r eik r r 4πɛ 0 r r, A ω = µ 0 V d3 x j(r eik r r 4π r r. (

23 2.4. MULTIPOLE RADIATED ANGULAR MOMENTUM Assuming a harmonic time dependence and expanding the sources we obtain the expressions [8] φ ω = C 4π(2l + 1 (4π 2 ɛ 0 ( iω l+1 Q lm ( ω 2 h (1 l (kry lm (θ,ϕ, (2.39 l,m (2l + 1!! A ω (r = µ 0 (2l + 1(l + 14π/l [ (4π 2 ( iω l+1 l,m (2l + 1!! 1 [ Qlm ( ω 2 + iωt lm ( ω 2 ] ] [h (1 l (kry l,l,m k ic ] l [ ] k l + 1 Q lm( ω 2 h (1 l (kry l,m im lm ( ω 2 h (1 l (kry l,l,m, (2.40 where C is a gauge constant; C = 0 correspond to solenoid gauge [8]. In the above expression c = 1 and the fact that the Fourier components of the form factors are considered has been emphasized by writing that they depend on ω 2. Furthermore, the spherical Hankel functions, i.e. h (1 l (kr are defined as in Ref. [8, 9, 19, 20]. These expressions again differ by a normalization factor of 4πi l to that of Ref. [14]. Setting C = 0 the electric field can be found from E ω = iωa ω, which yields E ω = µ 0 (2l + 1(l + 14π/l [ (4π 2 ( iω l+2 im lm ( ω 2 h (1 l (2l + 1!! 1 k l,m (kry l,l,m [ Qlm ( ω 2 + iωt lm ( ω 2 ] [h (1 l (kry l,l,m ]. (2.41 Comparing this with Eqs. (2.10 (2.11, we see that a E (l,m [ Q lm + k 2 T lm ] = ( iω [ Qlm ( ω 2 + iωt lm ( ω 2 ] (2.42 a M (l,m M lm ( ω 2, (2.43 The mutlipole expansion carried out in Ref. [14], for example, therefore implicitly contains the toroidal moments. The identification of the a E coefficients with the time dependent charge multipoles, which is done in Ref. [14] for small values of k, is incorrect. 2.4 Multipole Radiated Angular Momentum Returning to our TM and TE fields, i.e. equations (2.10 (2.11, it is a just a matter of calculating the product ɛ 0 Re{r (E B }/2 to find the angular momentum in terms of the multipoles. However, it is more elucidating to consider the special case of a source distribution described completely in terms of a E (l,m, or a M (l,m, multipoles of a specific 13

24 CHAPTER 2. MULTIPOLE EXPANSION l order. The m value may differ, though. Consider the radiation fields in this special case. For time harmonic fields, the two terms in the expression for the energy density u = ɛ 0 ( Eω 2 + c 2 B ω 2 ( are equal in the radiation zone [14]. Setting a M = 0 in Eqs. (2.10 (2.11, the energy within a spherical shell between r and r + dr can in the radiation zone be written as [14] du = µ 0dr 2k 2 a E(l,m a E (l,m dωy l,l,m Y l,l,m. (2.45 m,m From the orthogonality condition of the vector spherical harmonics, we find [5] du dr = µ 0 2k 2 a E (l,m 2. (2.46 m Similarly, using equation (2.7, the electromagnetic angular momentum density can be written as h = ɛ 0 2 Re{ r (E ω B ω } = 1 2µ 0 ω Re{ B ω(l B ω }. (2.47 The radiated angular momentum within a spherical shell is then given by [14] { dh = µ 0 2ωk 2 Re a E(l,m a E (l,m dω ( } L Y l,l,m Yl,l,m. (2.48 m,m Consequently, the standard result for the z component of the angular momentum is obtained as dh z dr = µ 0 2ωk 2 m a E (l,m 2. (2.49 For the special case of a single m value m dh z dr = m du ω dr, (2.50 reminiscent of the quantum mechanical case where photons carry hω units of energy while their angular momenta along the z axis are m h. However, since the relation between the a E coefficients with the sources is given by Eq. (2.42, this also relates the source distribution to the angular momentum radiated. In this sense, the multipoles are the sources of electromagnetic angular momentum. Indeed, the fact that in the case of transverse electric fields ωr B ω = 1 i r ( E ω = 1 i (r E ω = L E ω, (

25 2.5. FIELDS AND ANGULAR MOMENTUM FROM A TOROIDAL DIPOLE makes the OAM operator appear in the expression for angular momentum when written as h = ɛ 0 r (E B = ɛ 0 [E(r B B(r E]. Since the fields have been expanded in the eigenstates of the angular momentum operator, the multipoles, as coefficients multiplying the vector spherical harmonics, appear explicitly in the expression for angular momentum. They are then the relation between the source distribution and radiated angular momentum. 2.5 Fields and Angular Momentum from a Toroidal Dipole We shall now find the electric and magnetic fields from a toroidal dipole. In order to be able to compare our results with available ones, this section uses electrostatic units (ESU. The toroidal dipole moment is given by the expression T = 1 d 3 xr(r j 2r 2 j. ( c V This corresponds to the current distribution j = c ( [ T(tδ (3 (r ] = c ( (Tδ c 2 (Tδ, (2.53 while the charge density ρ = 0. The vector potential can be found from A(t,r = 1 c V d 3 x j(t,r r r, (2.54 where t denotes the retarded time, given by t = t r r. (2.55 c Inserting Eq. (2.53 into Eq. (2.54, the vector potential is, outside the origin, found to be given by A = 1 rc 2 T(t 1 cr 2 Ṫ(t 1 r 3 T(t + 1 r 3 c 2 ( T(t rr + 3 cr 4 (Ṫ(t rr + 3 r 5 (T(t rr, (2.56 in agreement with the results of Ref. [20]. In the units we are working with, the electric field is given by E = 1 Ȧ. (2.57 c 15

26 CHAPTER 2. MULTIPOLE EXPANSION The toroidal dipole selects a direction, just as does an electric dipole, which is along its symmetry axis. We fix a coordinate system such that the z axis is directed along this direction. Assuming a harmonic time dependence of the dipole, the expression for T is T(t = T 0 cosωtẑ. (2.58 This then gives the vector potential A = ω2 rc 2 T + ω cr 2 T 0 sinωt ẑ 1 r 3 T ω2 r 3 c 2 T 0 cosωt (ẑ rr 3ω cr 4 T 0 sinωt (ẑ rr + 3 r 5 T 0 cosωt (ẑ rr, (2.59 which in component form reads [ ω 2 A r = A ˆr = T 0 cosθ rc 2 cosωt + ω cr 2 T 0 sinωt 1 ] r 3 cosωt ω2 rc 2 T 0 cosωt cosθ 3ω cr 2 T 0 sinωt cosθ + 3 r 3 T 0 cosω cosθ [ = 2T 0 cosθ cosωt 1 r 3 ω ] sinωt cr 2, (2.60 [ ] ω 2 A θ = A ˆθ = T 0 sinθ ( 1r = T 0 sinθ [cosωt 3 ω2 rc 2 rc 2 cosωt + ω cr 2 sinωt 1 r 3 cosωt sinωt ω cr 2 ], (2.61 A ϕ = A ˆϕ = 0. (2.62 To find the magnetic field, consider the general expression for the curl of a vector field A = 1 [ ( Aϕ sinθ A ] θ ˆr+ 1 [ 1 A r r sinθ θ φ r sinθ ϕ ( ]ˆθ raϕ + 1 [ r r r (ra θ A ] r θ hence only one term need to be calculated, A = 1 [ r r (ra θ A ] r ˆϕ. (2.63 θ That is, B = B ϕ ˆϕ, where B ϕ = T [ ] 0 sinθ ω 2 r rc 2 cos(ωt ω3 c 3 sin(ωt The electric field, E = 1 c Ȧ = 1 c A r t ˆr 1 c. (2.64 A θ θ ˆθ, (2.65 ˆϕ, 16

27 2.5. FIELDS AND ANGULAR MOMENTUM FROM A TOROIDAL DIPOLE has the components ( ω 2 E r = 2T 0 cosθ c 2 r 2 cosωt + ω cr 3 sinωt, (2.66 [ ( ] ω 2 ω E θ = T 0 sinθ c 2 r 2 cosωt + sinωt cr 3 ω3 rc 3. (2.67 In our units S = 4π c (E B, hence S = c 4π ( E rb ϕ ˆθ + E θ B ϕˆr. (2.68 From the momentum density, i.e. S/c 2, the corresponding angular momentum density, in any zone, is found to be h = r 1 c 2 S = r c 2 S θ ˆϕ = h ϕ ˆϕ. (2.69 Performing the calculations gives and hence S r = ct 2 0 sin2 θ 4πr 2 S θ = ct 2 0 h ϕ = T 2 0 [ (ω 3 c 3 sinωt ω2 rc 2 cosωt 2 ω4 c 4 r 2 sin2 ωt + ω3 c 3 r 3 sinωt cosωt (2.70 [ ( ] sinθ cosθ ω 4 2πr c 4 r 3 cos2 ωt ω4 ω c 4 r 3 sin2 ωt 3 + sinωt c 3 r 4 ω5 c 5 r 2 cosωt, ] (2.71 [ ( ] sinθ cosθ ω 4 ω 3 2πc c 4 r 3 cos2ωt + sinωt c 3 r 4 ω5 c 5 r 2 cosωt. (2.72 The intensity in the radiation zone is given by I = lim r + dωs ˆrr 2. For the toroidal dipole of Eq. (2.53 we find that I = lim S r r 2 dω = dω c ω 6 r + 4π c 6 T 0 2 sin 2 ωt sin 2 θ = T 0 2ω6 sin 2 ωt 2π 1 4πc 5 dϕ (1 x 2 dx = c 5 (T 0ω 3 sinωt 2 = T 3c 5 t 3, (2.73 The angular momentum density in the radiation zone is given by h θ = T 2 0 ω5 sinθ cosθ 2πc 6 r 2 sinωt cosωt. (2.74, 17

28

29 3 A GENERAL EXPRESSION FOR ANGULAR MOMENTUM IN CLASSICAL ELECTRODYNAMICS The intensity (energy flux radiated through any shell of a concentric sphere enclosing the sources is constant. This energy is radiated out to infinity. The component of the product E B responsible for this is proportional to 1/r 2 at very large distances r from the source. There those components of the electric and the magnetic fields that fall off as 1/r are often called the radiation fields or the far fields. From the conservation laws in electrodynamics, just like the introduction of a field momentum, the concept of field angular momentum density has been introduced, r p field = r 1 c 2 S = ɛ 0r (E B = h. (3.1 The cross product of the radiation fields is directed radially, which, when inserted in Eq. (3.1 yields zero. However, angular momentum can also be radiated out to infinity. The field components that go as 1/r n for n 2, are called the near and intermediate fields. It is the very existence of these components that give rise to radiation of angular momentum. Expressions for the electric and magnetic fields are often found through the use of the potentials A and ϕ. They can in turn can be found through retarded integrals over the sources. However, it is possible to write down completely general solutions of Maxwell s equations for the electric and the magnetic fields in terms of such retarded integrals over the sources as well. The question is to what extent the explicit field components can be related to the radiated angular momentum. This question is more delicate then in the case of linear momentum since several, different combinations of the components of the fields can give a product whose moment is proportional to 1/r 2. 19

30 CHAPTER 3. GENERAL EXPRESSION FOR ANGULAR MOMENTUM 3.1 Arbitrary Source Distributions For an arbitrary given charge and current density, the electric and the magnetic fields can be written [27] B = µ [ 0 x j(t,r (r r 4π V d3 r r x j(t,r (r r ] c V d3 r r 2, (3.2 E = 1 [ x ρ(t,r (r r 4πɛ 0 V d3 r r x [j(t,r (r r ](r r c V d3 r r 4 + [ 1 x [j(t,r (r r ] (r r c V d3 r r c 2 x [ j(t,r (r r ] (r r ] V d3 r r 3, (3.3 where t = t r r c denotes the retarded time. We write, for convenience, B = B 1 + B 2, E = E 1 + E 2 + E 3 + E 4. (3.4 The identification of the terms with those in Eq. (3.2 and Eq. (3.3 should be clear. The field components B 2 and E 4 corresponds to the radiation fields, since they fall of as 1/r. The B 1, E 1, E 2 and E 3 fields are the near and intermediate fields. Although lengthy, a general expression for the electromagnetic angular momentum can be formed directly from its definition, i.e. Eq. (3.1, using the explicit field expressions Eq. (3.2 and Eq. (3.3. Writing the fields as in Eq. (3.4, and taking the product of each term separately, the results are obtained trivially so they will just be stated: h (11 = ɛ 0 r (E 1 B 1 ( 1 2 ([ = µ 0 r x ρ(t,r (r r ] [ 4π V d3 r r 3 x j(t,r (r r ] V d3 r r 3, (3.5 h (21 = ɛ 0 r (E 2 B 1 ( 1 2 ([ µ 0 = 4π c r x [j(t,r (r r ](r r ] [ V d3 r r 4 x j(t,r (r r V d3 r r 2 ], (3.6 h (31 = ɛ 0 r (E 3 B 1 20

31 3.1. ARBITRARY SOURCE DISTRIBUTIONS ( 1 2 ([ µ 0 = 4π c r x [j(t,r (r r ] (r r ] [ V d3 r r 4 x j(t,r (r r V d3 r r 3 ], (3.7 h (41 = ɛ 0 r (E 4 B 1 ( 1 2 ([ µ 0 = 4π c 2 r x [ j(t,r (r r ] (r r ] [ V d3 r r 3 x j(t,r (r r V d3 r r 3 ], (3.8 h (12 = ɛ 0 r (E 1 B 2 ( 1 2 ([ µ 0 = 4π c r x ρ(t,r (r r ] [ V d3 r r 3 x j(t,r (r r ] V d3 r r 2, (3.9 h (22 = ɛ 0 r (E 2 B 2 ( 1 2 ([ µ 0 = 4π c 2 r x [j(t,r (r r ](r r ] [ V d3 r r 4 x j(t,r (r r V d3 r r 2 ], (3.10 h (32 = ɛ 0 r (E 3 B 2 ( 1 2 ([ µ 0 = 4π c 2 r x [j(t,r (r r ] (r r ] [ V d3 r r 4 x j(t,r (r r V d3 r r 2 ], (3.11 h (42 = ɛ 0 r (E 4 B 2 ( 1 2 ([ µ 0 = 4π c 3 r x [ j(t,r (r r ] (r r ] [ V d3 r r 4 x j(t,r (r r ] V d3 r r 2. 21

32 CHAPTER 3. GENERAL EXPRESSION FOR ANGULAR MOMENTUM (3.12 The components can be rewritten in various way by using vector relations such as Eq. (3.24. However, not much insight is gained in the general case. The total expression h = h (11 + h (21 + h (31 + h (41 + h (12 + h (22 + h (32 + h (42, (3.13 is a general expression for the angular momentum of the electromagnetic field with explicit source dependencies. To the best of our knowlegde, this expression has not been given in the literature before. For analytical evaluations, there is a drawback using the equations (3.2 (3.3 since no Green function exists for the integrand. By writing multipoles in terms of distributions, the integrals of the angular momentum components can be evaluated. Next we shall see what the different terms mean in the case of electric dipole radiation. 3.2 Angular Momentum from an Electric Dipole In the multipole expansion approach of the previous Chapter, the electric dipole has a charge and current density in terms of distributions given by [8, 19] ρ d = (p(t δ(r, j d = ṗδ(r, (3.14 with p being the dipole moment. Indeed, we see that V d 3 xρ d r = = V V d 3 x [ r(p(t δ(r ] = d 3 xδ(rp i ˆx i = p(t V V d 3 xδ(rp i i r d 3 xδ(r = p(t, as it should. If a harmonic time dependent dipole is considered, such that p(t = pcosωt, the first term in the electric field can be calculated as 1 E 1 = x (r [ r cosωt p i i δ(r ] 4πɛ 0 V d3 r r 3 ( 1 cosωt = x δ(r p i (r r 4πɛ 0 V d3 i r r 3 ( 1 ωsinωt = x δ(r p i (r r i (r r 4πɛ 0 V d3 c r r 4 cosωt ˆx i r r 3 + 3cosωt (r r i (r r r r [ 5 1 ωsinωt ] (p ˆr = 4πɛ 0 cr 2 ˆr pcosωt r 3 + 3cosωt (p ˆr r 3 ˆr. (

33 3.2. ANGULAR MOMENTUM FROM AN ELECTRIC DIPOLE The following components of the electric field can be found immediately due to the delta function, E 2 = k sinωt 4πɛ 0 (p ˆr r 2 ˆr, (3.16 E 3 = k sinωt 4πɛ 0 r 2 [ p ˆr(p ˆr ], (3.17 E 4 = k2 cosωt (p ˆr ˆr. (3.18 4πɛ 0 r The total expression for the electric field reads E = 1 ( [k 2 (ˆr p ˆr cosωt cosωt + {3ˆr(p ˆr p} 4πɛ 0 r r 3 k ] sinωt r 2. (3.19 Writing this in complex Fourier form, where cosωt = Re { e i(kr ωt}, sinωt = Re { ie i(kr ωt}, the expression for the electric field becomes E ω = 1 [ k 2 (ˆr p ˆr 1 ( 1 4πɛ 0 r + (3ˆr(p ˆr p r 3 ik ] r 2 e ikr, (3.20 in agreement with the results in Ref. [14]. Similarly, the components of the magnetic field are found to be B 1 = µ 0 ( ωsinωt p ˆr 4π r 2, (3.21 B 2 = µ 0 4πc ( ω 2 cosωt p ˆr, (3.22 r which in complex notation becomes B ω = µ [ 0 iω (p ˆr 4π r = ck2 µ 0 (ˆr p 4π ] ω2 e ikr c r ] e ikr [ 1 1 ikr [ = ck2 µ 0 (p ˆr 1 i ] e ikr 4π kr r r, (3.23 also in agreement with Ref. [14]. With the use of the bac-cab rule a (b c = b(a c c(a b, (

34 CHAPTER 3. GENERAL EXPRESSION FOR ANGULAR MOMENTUM we can rewrite r (E B = E(r B B(r E. For the first term of the angular momentum density, i.e. h (11, we notice that the magnetic field is perpendicular to the radial direction. Hence, only one term in each angular momentum component need be calculated. We find h (11 = ɛ 0 [ B 1 (r E 1 ] = µ ( 0 ωsinωt [ (p ˆr ωsinωt ] (p ˆr (4π 2 r 2 + 2cosωt (p ˆr cr r 2 = ωµ [ ] 0 k sinωt (p ˆr(p ˆrsinωt (4π 2 r 3 + 2cosωt r 4 [ k sin 2 (ωt kr = ωµ 0 (4π 2 (p ˆr(p ˆr r2 r + 2sin(ωt krcos(ωt kr r 2 ], (3.25 h (21 = ɛ 0 [ B 1 (r E 2 ] = µ ( 0 ωsinωt [ (p ˆr k sinωt ] (p ˆr (4π 2 r 2 r = ωµ ( 0 k sin 2 ωt (p ˆr(p ˆr (4π 2 r 3. (3.26 Furthermore, from the expression for E 3 and E 4, equations (3.17 (3.18, it can be seen that these are also perpendicular to the radial direction. Therefore, the components h (31 = h (41 = 0. (3.27 Next, we continue with h (12 = ɛ 0 [B 2 (r E 1 ] = µ ( 0 ω 2 cosωt [ (p ˆr ωsinωt ] (p ˆr (4π 2 + 2cosωt (p ˆr c r cr r 2 [ ] = k2 cµ 0 k sinωt (p ˆr(p ˆrcosωt (4π 2 r 2 + 2cosωt r 3 [ ] = k2 cµ 0 k cos(kr ωtsin(kr ωt (p ˆr(p ˆr (4π 2 r 2 + 2cos2 (kr ωt r 3, (

35 3.2. ANGULAR MOMENTUM FROM AN ELECTRIC DIPOLE h (22 = ɛ 0 [B 2 (r E 2 ] = µ ( 0 ω 2 cosωt [ (p ˆr k sinωt ] (p ˆr (4π 2 c r r [ ] = ck2 µ 0 k sinωt (p ˆr(p ˆrcosωt (4π 2 r 2 = ck3 µ 0 ωtsin(kr ωt (p ˆr(p ˆrcos(kr (4π 2 r 2. (3.29 Again, due to the fact that the field components E 3, E 4 and B 2 are perpendicular to the radial direction, the last two angular momentum components vanish, h (32 = h (42 = 0. (3.30 The total angular momentum density is given by h = ωµ [ 0 (p ˆr(p ˆr 1 sin(kr ωtcos(kr ωt 8π2 r4 + k [ cos 2 r 3 (kr ωt sin 2 (kr ωt ] ] + k2 cos(kr ωtsin(kr ωt, (3.31 r2 which agrees with the results of Ref. [27]. It is interesting to note that the only components that are radiated out to infinity are found in h (22 and h (12. However, this is related to the fact that we are considering electric multipole radiation where the magnetic field is always perpendicular to the radial direction. Since the components E 3 and E 4 are orthogonal to this direction, the contribution from these components vanishes. For an electric dipole directed along a fixed axis, which we can choose to be the z axis, the occurrence of the factor (p ˆr(p ˆr in front of each angular momentum component means that they will all be accompanied by a sinθ cosθ dependence, in spherical coordinates. Thus, when integrated over a spherical surface, the total angular momentum will be zero. However, when integrated over a finite surface area, it can be different from zero. Furthermore, the radiated angular momentum has a time dependence that goes as sin(kr ωtcos(kr ωt, which will also vanish when integrated over a whole period in time. In other words, the time average of the radiated angular momentum is also zero, whereas the instantaneous angular momentum is not. 25

36

37 4 CONSERVATION LAWS FOR ORBITAL ANGULAR MOMENTUM In classical electrodynamics it is today standard nomenclature to say that a field with an e imϕ dependence, where ϕ is the azimuthal angle, is in a state of definite orbital angular momentum (OAM with corresponding OAM number (topological charge m. This is in analogy with the quantum mechanical case where such a state is an eigenstate of the z component of the OAM operator L z [24]. Allen et al. [1] talk about the separation of total angular momentum into spin and orbital parts by writing the expression for electromagnetic angular momentum in two separate terms proportional to the OAM m number and the helicity, respectively. Despite the fact that e imϕ is not an eigenfunction of the total OAM operator, only the z component of the photon angular momentum contributes to the total classical field angular momentum [14]. However, concerning theoretical aspects, it is important to separate the eigenstates of the total OAM operator and those of only the z component. The question is whether some specific conservation laws can be derived for pure OAM states. Corresponding versions of Poynting s theorem, the energy-momentum theorem, and the conservation law for angular momentum for OAM eigenstates will be derived in this chapter. The question of how to conveniently describe a field that is in a superposition of multiple OAM components can be resolved by Fourier transforming with respect to the azimuthal angle E m = 1 2π Ee imϕ dϕ. (4.1 2π 0 Cylindrical coordinates will be used since they are suitable for describing fields with an azimuthal symmetry. The relations between cylindrical and Cartesian coordinates are 27

38 CHAPTER 4. CONSERVATION LAWS FOR ORBITAL ANGULAR MOMENTUM given by { x = r cosϕ y = r sinϕ, { r = x 2 + y 2 ϕ = tan 1 y x. (4.2 We will assume a ϕ dependence, with ϕ time independent, of both fields and sources in the form since then E(t,r,ϕ,z = E m (t,r,ze imϕ, (4.3 m 1 2π 2π 0 dϕee imϕ = 1 2π 2π ˆx j dϕe j mϕ m m 0 (t,r,zei(m = 1 2π 2π ˆx j E j m (t,r,z m 0 dϕe i(m mϕ (4.4 (4.5 = ˆx j E j m(t,r,z = E m. (4.6 A similar approach is taken in Ref. [10], where the quantum mechanical analogy of the electromagnetic fields to the photon wave function is pushed to its full extent. In this article Elias [10] also expands the mode amplitudes, i.e. E m, into quantum eigenfunctions in ρ and z. We take a more classical approach and will not consider the dependencies of the fields in variables other than ϕ. In order to proceed we must, however, find the corresponding Maxwell equations in terms of these modes. 4.1 The Corresponding Maxwell Equations Orbital angular momentum is considered to be a non-local property of the field. Because of this, we shall start from the integral from of Maxwell s equations. However, it will be apparent that this will not be necessary in the following. From Eq. (1.1, S Hence V d 2 x ˆn E(t,r,ϕ,z = [ d 2 x ˆn S = m = V V m e imϕ E m ] = m V d 3 x [e imϕ E m ] d 3 x [ e imϕ E m + E m e imϕ] (4.7 d 3 x ρ ɛ 0 = m V d 3 x ρ m(t,r,z ɛ 0 e imϕ. (4.8 ( d 3 x [e imϕ E m ρ ] m + E m e imϕ = 0. (4.9 m ɛ 0 28

39 4.1. THE CORRESPONDING MAXWELL EQUATIONS Since the volume is arbitrary, the integrand must vanish: ( [e imϕ E m ρ m ɛ 0 m m + E m e imϕ ] = 0. (4.10 Multiplying by e imϕ /(2π and integrating over one period in ϕ gives [ ] 1 dϕe iϕ(m m 1 E m + E m dϕe imϕ e im ϕ 2π 2π By introducing the shorthand notation C(m,m = 1 2π we can rewrite Eq. (4.11 as [ 1 2π m m = ρ m ɛ 0. (4.11 dϕe imϕ e im ϕ, (4.12 ] dϕe iϕ(m m E m + E m C(m,m = ρ m ɛ 0. (4.13 Performing the same steps for the Maxwell equation for the divergence of the magnetic field, i.e. B = 0, yields the relation [ ] 1 dϕe iϕ(m m B m + B m C(m,m = 0. (4.14 2π It should now be apparent that the integral versions of Maxwell s equations need not to be considered since the volume or surface in question can be taken as arbitrary. However, in order to relate the different modes to each other, we must integrate over one complete period in ϕ. With this in mind, we rewrite the curl of the electric field [ E = m m E m e imϕ ] = m [ e imϕ E m E m e imϕ] = e imϕ B m. (4.15 m t Again multiplying by e imϕ /(2π and integrating over one period in ϕ gives [ ] 1 dϕe i(m mϕ E m E m C(m,m = B m. (4.16 2π t The corresponding law for the magnetic field is given by [ ] 1 dϕe i(m mϕ B m B m C(m,m 2π m = µ 0 j m + 1 c 2 E m t. (

40 CHAPTER 4. CONSERVATION LAWS FOR ORBITAL ANGULAR MOMENTUM Explicit Relations Between Modes Having expressed the fields as in Eq. (4.3, we find that the Cartesian derivatives of the OAM amplitudes are given by [ Em(t,r,z i ] = [ Em(t,r,z i ] r x r x = [ E i m (t,r,z ] x r r = [ E i m (t,r,z ] cosϕ, (4.18 r [ Em(t,r,z i ] = [ Em(t,r,z i ] r y r y = [ E i m (t,r,z ] y r r = [ E i m (t,r,z ] sinϕ, (4.19 r [ Em(t,r,z i ] = [ Em(t,r,z i ]. (4.20 z z This means that E m = x i Ei m = Ex m x + Ey m y + Ez m z = Ex m r cosϕ + Ey m r sinϕ + Ez m z. (4.21 By writing the trigonometric functions in terms of exponentials cosϕ = eiϕ + e iϕ, 2 sinϕ = eiϕ e iϕ, (4.22 2i we can calculate the integrals 1 2π dϕcosϕe i(m mϕ = 1 2π 0 4π ( dϕ e i[m (m 1]ϕ + e i[m (m+1]ϕ = 1 2 (δ m 1,m +δ m+1,m, ( π dϕsinϕe i(m mϕ = 1 2π 0 4πi ( dϕ e i[m (m 1]ϕ e i[m (m+1]ϕ = 1 2i (δ m,m 1 δ m,m+1. (4.24 This makes it possible to find the explicit relations between the different OAM modes in Eq. (4.13, where the first term is given by 1 dϕe iϕ(m m E m = 1 [ dϕe iϕ(m m E x ] m cosϕ + Ey m sinϕ + Ez m 2π 2π r r z = 1 Em x 2 r (δ m 1,m + δ m+1,m i E y m 2 r (δ m,m 1 δ m,m+1 + Ez m z δ m,m. (