axion electrodynamics in a waveguide
TRANSCRIPT
Physics Letters A 372 (2008) 2346–2349
www.elsevier.com/locate/pla
Axion electrodynamics in a waveguide
Adam Noble
Physics Department, Lancaster University and the Cockcroft Institute, Lancaster, LA1 4YB, UK
Received 21 September 2007; received in revised form 14 November 2007; accepted 19 November 2007
Available online 4 December 2007
Communicated by P.R. Holland
Abstract
The interaction of axion and electromagnetic waves is studied in the presence of a magnetic field threading a waveguide. This interaction,which vanishes in free space, is found to induce transverse magnetic waves with frequency spectra associated with transverse electric waves in theabsence of the axion.© 2007 Elsevier B.V. All rights reserved.
PACS: 03.50.De; 14.80.Mz
Keywords: Axion electrodynamics; Waveguides; Modified TM waves
1. Introduction
The fundamental structures underlying classical electromag-netism have recently generated considerable interest [1–3]. Inparticular, while Maxwell’s equations
(1)dF = 0, dH = J,
are well understood, the constitutive equations relating F andH have come under scrutiny. Here, F is the Faraday 2-form andH the excitation 2-form.
In order to close the system, Eq. (1) must be supplementedby a further relation among the fields. Although more generalrelations do occur in certain material media, it is common torestrict to the case where H is a local, linear functional of F :
Z :Λ2M → Λ2M,
(2)F �→ H =Z(F ).
In general, the pseudotensor Z has 36 components, though cer-tain symmetry requirements can reduce this significantly. Inparticular, if spacetime is assumed to be isotropic, the vacuum
E-mail address: [email protected].
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constitutive tensor has the form Z = ψ� + α̃1, or
(3)H = ψ � F + α̃F.
Since the Hodge map � takes forms to pseudoforms (and viceversa), it follows that ψ is a scalar field and α̃ is a pseudoscalar,that is, under a change of orientation ψ → ψ , α̃ → −α̃.
The scalar ψ , known as the dilaton field, relates to the per-mittivity and impermeability of free space. In ‘conventional’vacuum electrodynamics it is taken to be constant (ψ = 1 inunits such that ε0 = c = 1), while α̃, known as the axion fieldis usually assumed to vanish. However, there are a number offeatures which can make a nonzero axion elusive.
Firstly, if dα̃ = 0, the second of Eqs. (1) becomes d �F = J ,i.e. a constant axion does not contribute to Maxwell’s equations.(We have assumed here, as in the rest of this Letter, that ψ = 1.)
Also, it has been argued that even a varying axion field doesnot affect the geometrical optics limit of electrodynamics [4](though see also [5,6]). In order to detect the axion, we mustlook beyond geometrical optics. One possibility is to study ax-ion effects in a waveguide, where the boundaries have an im-portant effect on the propagation of electromagnetic waves.
In [7], the effects of certain media in a waveguide were ex-plored, and it was found that the axion (or Tellegen parameteras it is there called) could affect the dispersion relation of elec-
A. Noble / Physics Letters A 372 (2008) 2346–2349 2347
tromagnetic waves. By treating the axion as a dynamical field,rather than as a prescribed parameter, we can derive the preciseform of this modification.
In order to analyse this situation, we need to understand thebehaviour of the axion field. In Section 2 we postulate equationsto be satisfied by the axion and electromagnetic fields, and con-struct a scheme for perturbing about free fields. In Section 3 weconsider the geometric description of the waveguide, and theform of propagating fields within it. Section 4 looks at the formof the free fields, while Section 5 deals with the modification ofthe electromagnetic waves due to the axion.
Extensive use is made of exterior differential forms through-out this Letter. For further details, see for example [8].
2. Field equations for the Maxwell-axion system
For convenience, set α̃ = gα, with g a scalar coupling con-stant. Hereafter α (also a pseudoscalar field) will be referredto as the axion, rather than α̃. Then the field equations for theMaxwell-axion system arise from the action
(4)S[A,α] = −1
2
∫M
F ∧ �F + gαF ∧ F + dα ∧ �dα,
where F = dA and � is the Hodge map. This is the usual actionfor free electromagnetic and massless pseudoscalar fields, withan additional interaction term gαF ∧ F ∼ αE · B.
The field equations following from stationarity of this actionunder variations of A and α are, respectively,
(5)d � F + g dα ∧ F = 0,
(6)d � dα = g
2F ∧ F,
which must be supplemented by the equation
(7)dF = 0,
following from the property of the exterior derivative d2A = 0.In addition, this action leads to Neumann junction conditionsfor the axion field:
(8)τ ∧ [�dα] = 0,
where τ is the normal 1-form to a hypersurface, across which[G] is the discontinuity in a field G. In addition, discontinuitiesin the electromagnetic field must satisfy
(9)τ ∧ [F ] = 0.
The fields should also satisfy τ ∧ [H ] = j , with j the surfacecurrent at the discontinuity. If we assume discontinuities occurat perfectly conducting surfaces, we can take this condition todefine j , so it provides no further constraint. Note that the setof Eqs. (5)–(9) admit the solutions discussed in [9].
In the limit g → 0, the equations for F and α decouple,reducing to the standard Maxwell theory and a noninteract-ing scalar field. The success of conventional Maxwell theorysuggests that if the constitutive tensor contains an axion part,g must be very small, and can be used as a parameter in a per-turbation expansion.
Assume the fields may be expanded in powers of g accordingto the following ansatz:
F = F0 + gF1 + g2F2 +O(g3),
(10)α = gα1 + g2α2 +O(g3).
Substituting these ansätze into (5) and (6) leads to equationsfor each order in the expansion:
dF0 = 0, d � F0 = 0,
dF1 = 0, d � F1 = 0, d � dα1 = 1
2F0 ∧ F0,
(11)dF2 = 0, d � F2 + dα1 ∧ F0 = 0,
where only those equations to be utilised in the following havebeen displayed.
Before proceeding to solve these equations, consider theirstructure. The order g0 equations describe a standard electro-magnetic field F0, hereafter referred to as the background field.The order g1 equations represent a noninteracting electromag-netic field F1, and a pseudoscalar field α1 with the backgroundfield as source. It is only at order g2 that the axion induces anelectromagnetic field that differs from a standard vacuum solu-tion. In fact, F2 may be interpreted as a solution to Maxwell’sequation d � F2 = J2, with a source term J2 = −dα1 ∧ F0.
3. Propagating fields in a waveguide
A waveguide B bounds a 4-dimensional region W in space-time M . For the present purposes, we consider only cylindricalwaveguides at rest in an inertial frame in Minkowski spacetime;for an extension of the formalism to nonstraight waveguides ro-tating in a gravitational field, see [10,11].
In cylindrical polar coordinates, the Minkowski metric is
(12)g= −dt ⊗ dt + dz ⊗ dz + dρ ⊗ dρ + ρ2 dϕ ⊗ dϕ
and the waveguide bounding the region
(13)W = {p ∈ M: ρ � ρ0}is given by
(14)B = ∂W = {p ∈ W : ρ = ρ0},where ρ0 is the constant radius of the waveguide. The unit nor-mal to the waveguide is τ = dρ|B .
It is convenient to separate the metric into longitudinal andtransverse parts:
g� = −dt ⊗ dt + dz ⊗ dz,
(15)g⊥ = dρ ⊗ dρ + ρ2 dϕ ⊗ dϕ.
Associated with each of these metrics is a Hodge map, given bythe volume forms:
(16)#1 = dt ∧ dz, ∗1 = ρ dρ ∧ dϕ.
Vectors V ∈ T M satisfying g�(V ,−) = 0 will be calledtransverse vectors. The metric duals of transverse vectors, andexterior products of these, will be called transverse forms.
2348 A. Noble / Physics Letters A 372 (2008) 2346–2349
Exterior to the waveguide, assume that all fields vanish, withthe possible exception of F0. Then the junction conditions (8)and (9) imply that
(17)dρ ∧ �dα1|B = 0, dρ ∧ F1,2|B = 0.
We seek solutions to (11) with F0 representing a constantmagnetic field in the waveguide interior and F1,2 and α1 fieldspropagating along the waveguide. We end this section by defin-ing what we mean by ‘constant’ and ‘propagating’ fields.
For a scalar field κ ∈ Λ0M to be constant means dκ = 0.This definition cannot be extended to a 2-form F ∈ Λ2M , rep-resenting an electromagnetic field, since dF = 0 is satisfied byall Faraday 2-forms. Instead we take the constancy of a 2-formwith respect to a given frame {ea} to mean that its componentsin that frame are constant scalars:
(18)F = 1
2Fabe
a ∧ eb, dFab = 0.
For the present purposes, we take the frame {ea} = {dt, dz,
dρ,ρ dϕ}.By a propagating solution, we mean that the fields take the
form
(19)χ = χ̂ exp[i(kz − ωt)
]where ω and k are constant scalars and χ̂ does not depend on(t, z), though it may depend on (dt, dz) if χ /∈ Λ0M .
Two things should be noted about this definition of propagat-ing solutions. Firstly, it is rather restrictive, since it permits onlywaves of a single frequency. Secondly, it clearly involves com-plex fields, whereas the form of the action (4) requires the fieldsto be real. Both of these objections can be resolved by the ob-servation that the equations we use are linear in the propagatingfields, so we can represent wave packets by a superposition ofthe real and imaginary parts of (19) with a range of frequencies.
4. Low order solutions
4.1. Solving for F0
Consider first the constant field F0. We want this to representa magnetic field. The definition (18) together with (11) thenimplies:
(20)F0 = b0 ∗ 1, db0 = 0.
This is a magnetic field of constant magnitude b0, orientedalong the direction of propagation of the waveguide.
4.2. Solving for F1
To solve for F1 we assume that it can be split into transverseelectric (TE) and transverse magnetic (TM) modes of the form
FTE = (Ψ ∗1 + ΘTE) exp[i(kz − ωt)
],
(21)FTM = (Φ#1 + ΘTM) exp[i(kz − ωt)
],
where Ψ and Φ are functions of (ρ,ϕ) and (ΘTE,ΘTM) are 2-forms containing neither #1 nor ∗1. Note that permitted values
of ω and k will be different for the TE and the TM waves. Thejunction conditions (17) then imply that
dρ ∧ ΘTE|B = 0,
(22)Φ|B = 0, dρ ∧ ΘTM|B = 0.
Applying Eqs. (11) to FTE implies
(23)∗d ∗ dΨ + λ2Ψ Ψ = 0
where λ2Ψ := ω2 − k2 is an invariant measure of the TE wave
frequency. The boundary condition (22) on ΘTE is equivalentthe Neumann condition on Ψ :
(24)dρ ∧ ∗dΨ |B = 0.
FTE can then be fully reconstructed from solutions to (23).Similarly, applying Eqs. (11) to FTM implies
(25)∗d ∗ dΦ + λ2ΦΦ = 0,
where this time λ2Φ := ω2 − k2, for TM values of ω and k.
The first of the TM boundary conditions in (22) impliesdΦ|B ∼ τ . This implies that the second TM boundary condi-tion is necessarily satisfied. Thus the Dirichlet condition on Φ ,
(26)Φ|B = 0
is a sufficient condition for TM waves. Again, a solution for Φ
fully determines FTM.The solutions of (23) and (25) are well known: they are linear
combinations of the Bessel functions
Ψ mj = Jm
(λΨ m
jρ)eimϕ,
(27)Φmj = Jm
(λΦm
jρ)eimϕ,
where λΨ mj
ρ0 is the j th zero of J ′m, and λΦm
jρ0 is the j th zero
of Jm.
4.3. Solving for α1
Assume that the axion comprises two parts α1 = αb + αp ,with αb a nonpropagating (‘background’) solution and αp apropagating wave solution:
(28)d � dαb = 0, d � dαp = 0.
Under the assumption that dαb is a purely transverse 1-form,Eq. (28) and the boundary condition (17) constrain αb to be aconstant, which can be ignored since only dα enters the equa-tions.
The propagating part of the axion field αp is of the form:
(29)αp = γ exp[i(kz − ωt)
]with dγ a transverse 1-form.
Operating on the ansatz (29) with d � d yields
(30)∗d ∗ dγ + λ2γ γ = 0
where, as before, λ2 := ω2 − k2.
γ
A. Noble / Physics Letters A 372 (2008) 2346–2349 2349
Decomposing �dαp into parts proportional to ∗1 and #1, theboundary condition (17) becomes
(31)dρ ∧ ∗dγ |B = 0.
The Helmholtz equation for γ and the boundary conditionsit must satisfy are identical to those for Ψ considered above,and hence the solutions can be obtained from there:
(32)γ mj = Jm(λγ m
jρ)eimϕ,
where λγ mj
ρ0 is the j th zero of J ′m. Note that this implies
that the axion has the same spectrum of frequencies and wave-lengths as the TE modes of the electromagnetic field.
5. Axion induced modification of the electromagnetic wave
It is now possible to study the last of Eqs. (11) for the in-fluence of the axion on the propagation of the second orderelectromagnetic field F2. For any given solution α1, the sourceterm takes the form
J2 = −dα1 ∧ F0
(33)= (ib0γ ζ ∧ ∗1) exp[i(kz − ωt)
],
where ζ := ωdt − k dz. In order to fulfil d � F2 = J2, it is clearthat the (t, z)-dependence of F2 must match that of α1.
TE modes cannot satisfy this equation: this is to be expected,since the interaction term in the Lagrangian is proportional toF ∧ F ∼ b0FTM ∧ ∗1. Consider the TM waves. Inserting thedefinition (21) into Eqs. (11) for F2 leads to
(34)∗d ∗ dΦ + λ2γ Φ = b0λ
2γ γ,
along with the same Dirichlet boundary condition as before,ΦB = 0.
For a given solution γ to Eq. (30), the solution to (34) is
(35)Φ = −b0
2ρ
∂γ
∂ρ,
as can be verified by differentiating (30) with respect to ρ. It isclear that, since γ satisfies the Neumann boundary conditions,this solution vanishes on the boundary as required.
Such a solution gives for the second order electromagneticwave
F2 = −b0
2
[ρ
∂γ
∂ρ#1 + i
λ2γ
#ζ ∧ d
(ρ
∂γ
∂ρ
)]
(36)× exp[i(kz − ωt)
].
The longitudinal component of the electric field, ρ(∂γ /∂ρ),clearly differs in form from that of the free field F1. However,
since γ is a linear combination of the solutions (32) with arbi-trary coefficients, it may be difficult to distinguish this feature.However, as already noted, the axion has a frequency spectrumidentical to that of the free TE modes. Since the modified TMwaves acquire the same frequency and wavenumber as the ax-ion field driving them, a small amplitude TM wave with theusual TE frequency spectrum would be a clear indication of thepresence of the axion.
6. Conclusions
In this Letter, we have considered how an axion field prop-agating through a waveguide would interact with a constantmagnetic field oriented along the direction of propagation. Inthe absence of the waveguide, the axion would have no effect,since the electric field of the waves would be transverse to thedirection of propagation, so the interaction term αF ∧F wouldvanish.
We have shown that the presence of the waveguide allows aninteraction between the background magnetic field and a prop-agating axion field. This interaction induces small amplitudetransverse magnetic waves propagating with frequencies andwavelengths associated, in the absence of the axion, with TEmodes.
Acknowledgements
I would like to thank David Burton and Robin Tucker formany helpful discussions of this work, and an anonymous ref-eree for a number of helpful suggestions. This work was finan-cially supported by EPSRC.
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