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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 4 — Feb. 14, 2011
  • pp: 3742–3757
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Fresnel coefficients in materials with magnetic monopoles

J. Costa-Quintana and F. López-Aguilar  »View Author Affiliations


Optics Express, Vol. 19, Issue 4, pp. 3742-3757 (2011)
http://dx.doi.org/10.1364/OE.19.003742


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Abstract

Recent experiments have found entities in crystals whose behavior is equivalent to magnetic monopoles. In this paper, we explain some optical properties based on the reformulated “Maxwell” equations in material media in which there are equivalent magnetic charges. We calculate the coefficients of reflection and transmission of an electromagnetic wave in a plane interface between the vacuum and a medium with magnetic charges. These results can give a more extended vision of the properties of the materials with magnetic monopoles, since the phase and the amplitudes of the reflected and transmitted waves, differ with and without these magnetic entities.

© 2011 Optical Society of America

1. Introduction

The question of magnetic monopoles does not seem resolved and reappears periodically. Peregrinus [1

1. P. Peregrinus, The Letter of Petrus Peregrinus, translated by B. Arnold (McGraw Publishing Company, 1904).

] found, a long time ago, that breaking a magnet in half one had two magnets, each one with its north pole and south pole. Maxwell ruled out the existence of magnetic monopoles and postulated that always ∇ · B = 0. Later, Dirac argued [2

2. P. A. M. Dirac, “Quantized Singularities in the Electromagnetic Field,” Proc. R. Soc. Lond. A133, 60 (1931).

,3

3. P. A. M. Dirac, “The Theory of Magnetic Poles,” Phys. Rev. 74, 817–830 (1948). [CrossRef]

] that the existence of single magnetic monopoles would account for the quantization of the electric charges. Later, in the great unification theories the concept of magnetically charged particles reappears [4

4. G. ’t Hooft, “Magnetic monopoles in unified gauge theories,” Nucl. Phys. B79, 276–284 (1974). [CrossRef]

]. However, these monopoles have never been detected in the high energy range, and although some experiments seemed to announce some possible positive results [5

5. B. Cabrera, “First Results from a Superconductive Detector for Moving Magnetic Monopoles,” Phys. Rev. Lett. 48, 1378–1381 (1982). [CrossRef]

] no new experiment since then, to our knowledge, has been reproduced.

Recent experiments [6

6. C. Castelnovo, R. Moessner, and S. L. Sondhi, “Magnetic monopoles in spin ice,” Nature 451, 42–45 (2008). [CrossRef] [PubMed]

11

11. S. Ladak, D. E. Read, G. K. Perkins, L. F. Cohen, and W. R. Branford, “Direct observation of magnetic monopole defects in an artificial spin-ice system,” Nat. Phys. 6, 359–363 (2010). [CrossRef]

] have reawakened interest in the magnetic monopoles because of the possible behavior of determined magnetic structures which present similarities with the magnetic monopoles within some materials called spin-ices [8

8. S. T. Bramwell, S. R. Giblin, S. Calder, R. Aldus, D. Prabhakaran, and T. Fennell, “Measurement of the charge and current of magnetic monopoles in spin ice,” Nature 461, 956–960 (2009). [CrossRef] [PubMed]

12

12. S. Sondhi, “Wien route to monopoles,” Nature 461, 888–889 (2009). [CrossRef] [PubMed]

], or also as an image charge of an electric charge in topological insulators [7

7. X.-L. Qi, R. Li, J. Zang, and S.-C. Zhang, “Inducing a Magnetic Monopole with Topological Surface States,” Science 323, 1184–1187 (2009). [CrossRef] [PubMed]

]. While we wait new experimental verifications, these achievements [8

8. S. T. Bramwell, S. R. Giblin, S. Calder, R. Aldus, D. Prabhakaran, and T. Fennell, “Measurement of the charge and current of magnetic monopoles in spin ice,” Nature 461, 956–960 (2009). [CrossRef] [PubMed]

11

11. S. Ladak, D. E. Read, G. K. Perkins, L. F. Cohen, and W. R. Branford, “Direct observation of magnetic monopole defects in an artificial spin-ice system,” Nat. Phys. 6, 359–363 (2010). [CrossRef]

] seem to have more possibilities of being conclusive about the manifestation of determined magnetic structures as magnetic monopoles than those carried out in 1982 [5

5. B. Cabrera, “First Results from a Superconductive Detector for Moving Magnetic Monopoles,” Phys. Rev. Lett. 48, 1378–1381 (1982). [CrossRef]

]. However, the magnetic structures, which present this behavior equivalent to those of magnetic charges are not based on elementary particles but composite entities appearing in determined magnetic crystals [6

6. C. Castelnovo, R. Moessner, and S. L. Sondhi, “Magnetic monopoles in spin ice,” Nature 451, 42–45 (2008). [CrossRef] [PubMed]

, 8

8. S. T. Bramwell, S. R. Giblin, S. Calder, R. Aldus, D. Prabhakaran, and T. Fennell, “Measurement of the charge and current of magnetic monopoles in spin ice,” Nature 461, 956–960 (2009). [CrossRef] [PubMed]

11

11. S. Ladak, D. E. Read, G. K. Perkins, L. F. Cohen, and W. R. Branford, “Direct observation of magnetic monopole defects in an artificial spin-ice system,” Nat. Phys. 6, 359–363 (2010). [CrossRef]

]. In these crystals, determined spin flips produced in ions which serve for binding two contiguous tetrahedral molecular structures can have similar effect concerning the magnetic field creation to that of two magnetic charges of opposite signs [6

6. C. Castelnovo, R. Moessner, and S. L. Sondhi, “Magnetic monopoles in spin ice,” Nature 451, 42–45 (2008). [CrossRef] [PubMed]

, 8

8. S. T. Bramwell, S. R. Giblin, S. Calder, R. Aldus, D. Prabhakaran, and T. Fennell, “Measurement of the charge and current of magnetic monopoles in spin ice,” Nature 461, 956–960 (2009). [CrossRef] [PubMed]

, 9

9. D. J. P. Morris, D. A. Tennant, S. A. Grigera, B. Klemke, C. Castelnovo, R. Moessner, C. Czternasty, M. Meissner, K. C. Rule, J.-U. Hoffmann, K. Kiefer, S. Gerischer, D. Slobinsky, and R. S. Perry, “Dirac Strings and Magnetic Monopoles in the Spin Ice Dy2Ti2O7,” Science 326, 411–414 (2009). [CrossRef] [PubMed]

, 11

11. S. Ladak, D. E. Read, G. K. Perkins, L. F. Cohen, and W. R. Branford, “Direct observation of magnetic monopole defects in an artificial spin-ice system,” Nat. Phys. 6, 359–363 (2010). [CrossRef]

, 12

12. S. Sondhi, “Wien route to monopoles,” Nature 461, 888–889 (2009). [CrossRef] [PubMed]

]. Then, the action of an external magnetic field can compete with the attractive interaction between the two charges of the microscopic magnetic dipole leaving free its two charges, which can move inside the solid as the electric charges are moved in an electrolyte [6

6. C. Castelnovo, R. Moessner, and S. L. Sondhi, “Magnetic monopoles in spin ice,” Nature 451, 42–45 (2008). [CrossRef] [PubMed]

, 8

8. S. T. Bramwell, S. R. Giblin, S. Calder, R. Aldus, D. Prabhakaran, and T. Fennell, “Measurement of the charge and current of magnetic monopoles in spin ice,” Nature 461, 956–960 (2009). [CrossRef] [PubMed]

, 12

12. S. Sondhi, “Wien route to monopoles,” Nature 461, 888–889 (2009). [CrossRef] [PubMed]

, 13

13. L. J. Onsager, “Deviations from Ohm’s Law in Weak Electrolytes,” J. Chem. Phys. 2, 599–615 (1934). [CrossRef]

].

This work is a theoretical analysis based on the extended electromagnetism equations for the case in which the magnetic monopoles and electric charges coexist within a physical system. These extended equations, whose description in vacuum is perfectly known [14

14. See for example, J. D. Jackson, Classical Electrodynamics, third edition (John Wiley & Sons, Inc., 1999).

16

16. J. Costa-Quintana and F. Lopez-Aguilar, “Extended classical electrodynamics with magnetic monopoles,” Far East J. Mech. Eng. Phys. 1, 19–56 (2010).

], are applied to the cases in which the coexistence among electric and magnetic charges occurs within the matter [16

16. J. Costa-Quintana and F. Lopez-Aguilar, “Extended classical electrodynamics with magnetic monopoles,” Far East J. Mech. Eng. Phys. 1, 19–56 (2010).

, 17

17. J. Costa-Quintana and F. Lopez-Aguilar, “Propagation of electromagnetic waves in material media with magnetic monopoles,” Prog. Electromagn. Res. 110, 267–295 (2010). [CrossRef]

]. One of our objectives being the attempt to give a guide of theoretical properties deduced with field equations of the extended electromagnetism in order to induce experiments able to detect properties of both the spin-ices and new materials in which the magnetic monopoles are possible. Within this general philosophy, we study in this work the propagation of electromagnetic waves in different conducting media where unconfined monopoles can exist. In a first step, we calculate the propagation vector (K) that in general is a complex vector, as in a conducting material. We make a complementary analysis of Classical Physics to determine the electromagnetic transmittance and reflectance of an incident wave in the separation surfaces of two different media, where one of them can be a spin-ice material. Using the boundary conditions we calculate the magnitudes and polarizations of the different fields E and B, as well as the relationships between the incident electromagnetic wave and the reflected/transmitted ones, i.e., we determine the so-called Fresnel coefficients. The measurement of the reflection coefficient of an electromagnetic wave incident on a material with magnetic monopoles can be decisive in confirming the existence of such magnetic monopoles as well as their optical properties.

2. Fields equations with monopoles

With the assumption of existence of the magnetic monopoles the modified Maxwell equations in vacuum are [14

14. See for example, J. D. Jackson, Classical Electrodynamics, third edition (John Wiley & Sons, Inc., 1999).

17

17. J. Costa-Quintana and F. Lopez-Aguilar, “Propagation of electromagnetic waves in material media with magnetic monopoles,” Prog. Electromagn. Res. 110, 267–295 (2010). [CrossRef]

]:
E=ρeɛ0,×E=KJmBt,
(1)
B=Kρm,×B=μ0Je+μ0ɛ0Et,
(2)
where ρe is the electric charge density, ρm the magnetic monopole density, Je the electric current density and Jm the magnetic current density. The constant K can have any value since it defines the unit of magnetic charge. Obviously for K = 0 we have the standard equations of electromagnetism.

The Lorentz force for a point charge with electric (q) and magnetic (g) value can be written as:
F=q(E+v×B)+Kμ0g(Bvc2×E).
(3)

The above “Maxwell” equations can be expressed in a more unified way if one defines
κɛ0cK=Kμ0c,Ω(0110),
(4)
the electromagnetic charge and current density and the electromagnetic field as
Q(qκg),ρ(ρeκρm),J(JeκJm),G(EcB).
(5)
Note that an electromagnetic charge Q (sometimes called dyon [18

18. Y.M. Shnir, Magnetic monopoles, (Springer-Verlag, 2005).

]) has two components, an electric, q, and a magnetic, κg, charge. Similarly, the other magnitudes have two components: one in a subspace that is “electrical”, and another that is “magnetic”.

Then, the Maxwell equations (1) and (2) can be written as follows
G=1ɛ0ρ,×G=1ɛ0cΩJ+1ctΩG,
(6)
and the Lorentz force takes the form:
F=(q,κg)(𝟙vc×Ω)G.
(7)

2.1. Duality transformation

In Eq. (5), the charge, current and G-vector are presented in an electromagnetic double space where the up (down) subspace corresponds to the electric (magnetic) component. This allows us to make a representation of both the field sources (charges and currents) and the fields in the complex plane so that Eq. (5) can be re-written as
Qq+iκg=|Q|eiζ=|Q|(cosζ+isinζ)ρρe+iκρm,JJe+iκJm,GE+icB.
(8)
In this complex representation, the Maxwell equation (6) are given by
G=1ɛ0ρ,×G=ic(Jɛ0+Gt),
(9)
and the Lorentz force takes the form
F=Re[Q*(1ivc×)G].
(10)
In this notation, the particles are dyons whose charge is defined by the ζ-angle and its module |Q|. Thus, the electron, negatively charged, has a value of ζ = π, the proton is a dyon with ζ = 0 and for a magnetic monopole ζ = ±π/2. It must be remarked that Eqs. (6) and (7) are totally equivalent to Eqs. (9) and (10). However, in these latter equations of the complex representation, the explanation and the meaning of the duality transformation [14

14. See for example, J. D. Jackson, Classical Electrodynamics, third edition (John Wiley & Sons, Inc., 1999).

, 15

15. K. A. Milton, “Theoretical and experimental status of magnetic monopoles,” Rep. Prog. Phys. 69, 1637–1711 (2006). [CrossRef]

] is easier and clearer. The duality transformation [14

14. See for example, J. D. Jackson, Classical Electrodynamics, third edition (John Wiley & Sons, Inc., 1999).

, 15

15. K. A. Milton, “Theoretical and experimental status of magnetic monopoles,” Rep. Prog. Phys. 69, 1637–1711 (2006). [CrossRef]

] consists of a rotation of an angle ζ0 in the dyonic complex plane; the charges represented as Q ≡ |Q| exp() are transformed in Q′ ≡ Qexp(0) = |Q| exp[i(ζ + ζ0)], and the electromagnetic field components, defined as Gx,y,zEx,y,z + icBx,y,z, are transformed in the same way, G′x,y,z = Gx,y,z exp(0). Then, the Maxwell equations are invariant, that is, the equations of the primed field (G′) are the same as Eq. (9) with the primed sources present [14

14. See for example, J. D. Jackson, Classical Electrodynamics, third edition (John Wiley & Sons, Inc., 1999).

], since this rotation implies a multiplication of both sides of the equality by the same phase [exp(0)]. Furthermore, the Lorentz force is the same before and after the transformation, because the product ρ*G does not suffer any change by multiplying by exp(−iζ0) exp(0).

This duality transformation has a well defined physical meaning. Namely, when the charges are in the real axis of the complex plane, the standard Maxwell equations explain the electromagnetic phenomenology. And, if one rotates all the charges the same angle in the complex plane, the transformed equations with the rotated charges give the same results as the standard Maxwell equations. This theoretical assertion is coherent with the recent experimental study of Wien effect [8

8. S. T. Bramwell, S. R. Giblin, S. Calder, R. Aldus, D. Prabhakaran, and T. Fennell, “Measurement of the charge and current of magnetic monopoles in spin ice,” Nature 461, 956–960 (2009). [CrossRef] [PubMed]

]. This study proves that the conductivity measurement of the magnetic monopoles within the spin-ice gives the same conductance as the ionic (electric) charges dissolved in an electrolyte using the Onsager theory [12

12. S. Sondhi, “Wien route to monopoles,” Nature 461, 888–889 (2009). [CrossRef] [PubMed]

, 13

13. L. J. Onsager, “Deviations from Ohm’s Law in Weak Electrolytes,” J. Chem. Phys. 2, 599–615 (1934). [CrossRef]

]. In the Onsager equations of this theory, the only change that is necessary in order to interpret the conductance results in the spin-ices is the substitution of the electric charges and the universal electric constant for the corresponding magnetic ones [8

8. S. T. Bramwell, S. R. Giblin, S. Calder, R. Aldus, D. Prabhakaran, and T. Fennell, “Measurement of the charge and current of magnetic monopoles in spin ice,” Nature 461, 956–960 (2009). [CrossRef] [PubMed]

, 12

12. S. Sondhi, “Wien route to monopoles,” Nature 461, 888–889 (2009). [CrossRef] [PubMed]

].

A consequence of the invariance under a duality transformation is the conventional nature of the assignation of a determined dyon charge to the elementary particles, such as electrons protons, etc. If, by convention, it is established that the electron has a negative charge (dyon with ζ = π), the proton should be a dyon with ζ = 0. A positive charge (ζ = 0) or a magnetic monopole (ζ = ±π /2) could have been assigned to the electron with identical legitimacy but the ζ-angle of the other particles should be consistently changed. This is equivalent to say that if the electron charge is located in a dyonic line (points with the same value of ζ) in the complex plane, then the proton charge should be located in the same line in the symmetric point with respect the origin.

The novelty in our theoretical analysis is the formulation of an extended electromagnetism within the matter in which there is coexistence of magnetic charges, which are ζ = π/2-dyons, with electric charges whose ζ-phases are either 0 or π. This analysis has been encouraged by the recent experimental analysis [8

8. S. T. Bramwell, S. R. Giblin, S. Calder, R. Aldus, D. Prabhakaran, and T. Fennell, “Measurement of the charge and current of magnetic monopoles in spin ice,” Nature 461, 956–960 (2009). [CrossRef] [PubMed]

, 9

9. D. J. P. Morris, D. A. Tennant, S. A. Grigera, B. Klemke, C. Castelnovo, R. Moessner, C. Czternasty, M. Meissner, K. C. Rule, J.-U. Hoffmann, K. Kiefer, S. Gerischer, D. Slobinsky, and R. S. Perry, “Dirac Strings and Magnetic Monopoles in the Spin Ice Dy2Ti2O7,” Science 326, 411–414 (2009). [CrossRef] [PubMed]

, 11

11. S. Ladak, D. E. Read, G. K. Perkins, L. F. Cohen, and W. R. Branford, “Direct observation of magnetic monopole defects in an artificial spin-ice system,” Nat. Phys. 6, 359–363 (2010). [CrossRef]

], that have detected this coexistence of dyons of different lines of the complex plane in both the spin-ices and topological insulators [7

7. X.-L. Qi, R. Li, J. Zang, and S.-C. Zhang, “Inducing a Magnetic Monopole with Topological Surface States,” Science 323, 1184–1187 (2009). [CrossRef] [PubMed]

].

2.2. Field equations in matter

If the modified Maxwell equations in vacuum are well known, the corresponding ones within the matter are less known, since from the treatment of Dirac’s monopole arguments, the appearance of magnetic monopoles was expected to occur in elementary particles coming either from the prospection of a new range of high energies or from the research of particle radiation of the astrophysical exterior space. However, at the present time, entities whose behavior is as if they were magnetic monopoles have been experimentally detected only in the spin-ices and topological insulators. This has as a consequence that while the analysis of gauge physics of Dirac’s magnetic monopoles have already been analyzed (see for instance, a recent review [15

15. K. A. Milton, “Theoretical and experimental status of magnetic monopoles,” Rep. Prog. Phys. 69, 1637–1711 (2006). [CrossRef]

]), the analysis of the existence of these points with ∇ · B ≠ 0 in solid is not sufficiently treated being far from constituting a closed theoretical matter.

One of the key objects in the definition of the electrodynamics within the matter is the dipole moment concept, whose extended definition when there are magnetic charges (or equivalent magnetic charges) can be formulated by as follows:
P(PeκPm)1Δ𝒱Δ𝒱rρd3r,M(MeκMm)12Δ𝒱Δ𝒱r×Jd3r.
(11)
Here, Pe (Pm) is the electric (magnetic) polarization due to a split in the gravity centers of the electric (magnetic) charges, and Me (Mm) is the magnetization due to the electric (magnetic) currents. We call P split-charge polarization and M kinetic polarization.

Then the first Maxwell equations in material media are [16

16. J. Costa-Quintana and F. Lopez-Aguilar, “Extended classical electrodynamics with magnetic monopoles,” Far East J. Mech. Eng. Phys. 1, 19–56 (2010).

]
d=ρ,dɛ0G+P,
(12)
where we have defined the corresponding extended concept of the displacement vector. And the second Maxwell equations are
×h=Ω(J+dt),h1μ0cGΩM.
(13)

It must be remembered the coherence with the standard classical electrodynamics, if there are not magnetic monopoles (κ = 0), we have
d=(Dɛ0cB),h=(ɛ0cEH).
(14)

2.3. Linear responses of the materials

The condensed matter can response in different ways before the electromagnetic fields. This response depends on the characteristic properties of the material medium. Therefore, it is necessary to find relationships between the fields and the responses induced in the material. Fundamentally, these responses are: (i) the movement of charges, which, in this case, can be magnetic and electric, (ii) changes of the positions of the charges depending on their signs provoked by the interactions of the different fields, which constitute the split-charge polarization and (iii) changes of the localized currents, or the angular momentum of the particles also due to actions of the E and B, which constitute the so-called kinetic polarization.

Conductivity

As Bramwell et al. [8

8. S. T. Bramwell, S. R. Giblin, S. Calder, R. Aldus, D. Prabhakaran, and T. Fennell, “Measurement of the charge and current of magnetic monopoles in spin ice,” Nature 461, 956–960 (2009). [CrossRef] [PubMed]

] have recently shown, the equivalent entities to magnetic monopoles present under the magnetic field action identical behavior to that of the electric charges of an electrolyte under the electric field. This totally agrees with the duality transformation principle above explained. Therefore, Ohm’s law for magnetic charges can be formulated in perfect symmetry with the standard Ohm’s law of the electric ones. As a consequence, one can write in the following matrix form the current density [16

16. J. Costa-Quintana and F. Lopez-Aguilar, “Extended classical electrodynamics with magnetic monopoles,” Far East J. Mech. Eng. Phys. 1, 19–56 (2010).

]
J=σG
(15)
that is Ohm’s law. The conductivity is a matrix defined by
σσΘ,
(16)
where σ, in linear approximation, is a constant characteristic of each material media and the Θ matrix are
Θ(cos2ζcosζsinζcosζsinζsin2ζ).
(17)
If there is more than one class of particles (particles with a different value of tan ζ), then, the conductivity becomes the sum of all contributions, i.e.,
σ=pσpΘp,
(18)
where ζp is the electromagnetic angle in the dyonic complex plane for each class of particles. In the specific case of coexistence of pure electric and magnetic charges (i.e., dyons exclusively located either in the real axis or imaginary axis of the complex plane), the conductivity is:
σ=(σ00σ¯).
(19)

Split-charge susceptibility

Similarly to the conductivity case, we can establish the split-charge susceptibility [16

16. J. Costa-Quintana and F. Lopez-Aguilar, “Extended classical electrodynamics with magnetic monopoles,” Far East J. Mech. Eng. Phys. 1, 19–56 (2010).

], χ, relationship between the density of split-charge dipole moment P and the electromagnetic field, G:
P=ɛ0χΘG,
(20)
therefore
dɛ0G+P=ɛ0(𝟙+χΘ)G=ɛG,
(21)
and the permittivity matrix is
ɛɛ0(𝟙+χΘ)=ɛ0(1+χcos2ζχcosζsinζχcosζsinζ1+χsin2ζ).
(22)
If there are no magnetic monopoles
ɛ=ɛ0(1+χe001),
(23)
which obviously reduces to the expression of standard electromagnetism (χe = χ, is the electric susceptibility). If there are only magnetic monopoles, we have:
ɛ=ɛ0(1001+χ¯e),
(24)
where, χ̄e is the magnetic charge susceptibility equivalent to the electric susceptibility. If there are several kinds of particles of the same class (different particles but with the same value of tan ζ) one needs to modify only the value of χ. With several class of particles it is also true, d = ɛG, and then, we have the following expression for ɛ⃡:
ɛ=ɛ0(1+Σpχpcos2ζpΣpχpcosζpsinζpΣpχpcosζpsinζp1+Σpχpsin2ζp)=ɛ0p(𝟙+χpΘp).
(25)

Kinetic susceptibility

In magnetic materials, the standard electrodynamic establishes that the induced magnetic moment is proportional to B. This B produces a magnetic force (v × B) which is able to change the current and the magnetic polarization. The extension of this concept to the dyonic currents requires the consideration of the Lorentz force proportional to the velocity,
Fv=v×(qBκcgE).
The changes of the kinetic polarization become:
(MeMm)=nα(qg)(qBκcgE),
(26)
where α′ is a response function which depends on the concrete material. Because this response function depends on the speed, we call it as kinetic polarizability of the material. In linear approach [16

16. J. Costa-Quintana and F. Lopez-Aguilar, “Extended classical electrodynamics with magnetic monopoles,” Far East J. Mech. Eng. Phys. 1, 19–56 (2010).

] M is:
M=ξμ0c(cosζsinζcos2ζsin2ζcosζsinζ)G.
(27)
We can define the permeability matrix, μ⃡,
h1μ0cGΩM=1μ0c[𝟙ξ(sin2ζcosζsinζcosζsinζcos2ζ)]G1cμ1G.
(28)

By matrix inversion
μ=μ01ξ(𝟙ξΘ),
(29)
and
G=cμh.
(30)

Without magnetic monopoles (ζ = 0)
μ=μ0(1001+χm),
(31)
that is the classical expression, with χm = ξ/(1 – ξ) the magnetic susceptibility. And in the case in which one has only magnetic monopoles (ζ = π/2)
μ=μ0(1+χ¯m001).
(32)

If there are several kinds of particles of the same class one need to modify only the value of ξ. If one considers several classes of particles, the expression G = c μh, continues being valid, and we have the following expression
μ=μ0(1Σpξpsin2ζpΣpξpcosζpsinζpΣpξpcosζpsinζp1Σpξpcos2ζp)1=μ0det|1ΣpξpΘp|(1pξpΘp).
(33)

3. Plane waves

For a monochromatic plane wave, the electromagnetic field is
h=(fb)exp[i(Krωt)].
(34)
Henceforth, the argument line of this work consists of the inclusion of this trial electromagnetic wave [Eq. (34)] in the Maxwell equations in order to obtain the electric and magnetic field (f and b) in matter, and the propagation K-wave vector versus the linear responses of material, σ⃡, μ⃡ and ɛ⃡. This is carried out by considering the continuity conditions of the fields yielded by the extended field equations.

In a linear medium without free charge, the Maxwell equation (12) is satisfied, and with Eqs. (21), (30) one obtains the following
0=d=cɛμhcɛμ(fb)exp[i(Krωt)].
(35)
this implies,
ɛμ(KfKb)=0,
(36)
if we can invert the matrix ɛμ⃡ (i.e., det|ɛμ⃡| ≠ 0), then we have the transversality condition of the fields
Kf=Kb=0.
(37)
where it must be remembered that f and b are the electric and magnetic flavors of the h-vector

The curl equation (13) can be written
×h=cΩ(σ+ɛt)μh,
that for the above given plane wave reads
K×(fb)+(SRUT)(fb)=0,
(38)
where the definition of the propagation matrix is
(UTSR)c(ωɛ+iσ)μ.
(39)
Writing the x, y, z-components the Eq. (38) reads
0=[(SKzKyKzSKxKyKxS)R(100010001)U(100010001)(TKzKyKzTKxKyKxT)](fxfyfzbxbybz).
(40)
This equation can have a solution different from the trivial (fx,y,z = bx,y,z = 0) if the corresponding determinant vanishes, i.e.,
0=det|(SKzKyKzSKxKyKxS)(TKzKyKzTKxKyKxT)+RU(100010001)|.
(41)
In a general case, the K-vector should have the k + il-structure, where k is really the propagation vector and l is the electromagnetic wave extinction vector, present in a conductor media. Therefore, having in mind the freedom for choosing a coordinate system, the general expression of the K-wave vector can be written as follows:
K=(kx+ilx)ex+kyey=Kxex+kyey,
(42)
where kx, lx i ky are real numbers. We just choose the vector ex in the direction of the imaginary part of K and then we make a rotation until K has no z component (i.e., Kz = 0 and the x-component of K is complex). Then Eq. (41) with the K-wave vector of Eq. (42) becomes
0=det|(Δky2KxkyδkyKxkyΔKx2δKxδkyδKxΔKx2ky2)|=Δ[(ΔK2)2+δ2K2],
(43)
where Δ ≡ RUST, δST and K2Kx2+ky2. If Δ ≠ 0, then
K2±iδKΔ=0K=±[iδ2±12(4Δδ2)1/2],
(44)
the combination of signs gives four possible solutions.

On the other hand, if there is only one class of dyons, then the result is the total symmetry between the propagation of the electromagnetic waves when there is only either electric or magnetic conductivity, since
K2=ɛμω2[1+iσωɛ],
(48)
which is the standard expression for an electric conductor, and
K2=ɛ¯μ¯ω2[1+iσ¯ωɛ¯],
(49)
which will be the expression for an magnetic conductor. This symmetry between Eqs. (48) and (49) is coherent with that of Bramwell et al’s experiments [8

8. S. T. Bramwell, S. R. Giblin, S. Calder, R. Aldus, D. Prabhakaran, and T. Fennell, “Measurement of the charge and current of magnetic monopoles in spin ice,” Nature 461, 956–960 (2009). [CrossRef] [PubMed]

] about the conductivity of magnetic charges, which also present total symmetry with Onsager’s theory [13

13. L. J. Onsager, “Deviations from Ohm’s Law in Weak Electrolytes,” J. Chem. Phys. 2, 599–615 (1934). [CrossRef]

] of the movement of electric charges in electrolytes. Actually, the behavior of an electromagnetic wave incident in an electric conductor is similar to the incidence in an electric insulator which is, at the same time, a magnetic charge conductor.

4. Fresnel coefficients

In this section, we carry out the analysis of the electric and magnetic fields when an electromagnetic wave is incident on a separation surface of two different media, and the relationships between the fields of the incident wave with those of the transmitted and reflected ones. Obviously, the interest is centered when one of these material media contains dyons whose charge is out of the real axis of the dyonic complex plane. We consider a monochromatic wave such as
h=h0exp[i(k0rωt)],
(50)
with k0 real, that propagates from vacuum to a medium. We consider that the surface separating the two media coincides with the plane x = 0. Obviously, such as it occurs in the standard electromagnetism, there will be a reflected wave h″ with k″ real, and a transmitted wave h′ which in a general case, considering the possibility of existing electric and magnetic charge conductivities, K′ is complex. To satisfy the boundary conditions of the d and h-fields [whose deduction can be extended from the standard electrodynamics but considering Eqs. (1), (2), (12) and (13)] at any point of the separation surface between media, must be
ω=ω=ω,ky=ky=ky,kz=kz=Kz.
(51)
Since the medium in which there is the initial traveling wave is the vacuum, then k0y = ky = ky and k0z = kz = kz are real. Besides, we can rotate the coordinates system so that 0 = k0z = kz = kz. In the medium were h′ propagates K′ may be complex, but as kz = 0 and ky are real, it has imaginary component only in the x-direction, i.e., we can always choose a system of coordinates in which
K=(kx+ilx)ex+kyeyKxex+kyey,
(42)
where kx, lx i ky are real and Kx complex. Applying the boundary conditions k0y = k0 sin θ = ky, where θ is the angle of incidence, the wave vector magnitude is
K2=KK=Kx2+k0y2Kx=(K2k02sin2θ)1/2,
(52)
and with the help of (45), the wave vector K′ = (Kx, k0 sin θ, 0) is fully determined.

4.1. TE polarization

First consider that f0 has only z-component, fz0, i.e., f0 is perpendicular to the plane of incidence, and therefore, parallel to the separation plane between the vacuum and the material. It is called s-polarization or transverse electric (TE) polarization.

From the curl equation (38), for a plane wave
K×f=Sf+Rbb=1R(S+K×)f
(53)
that can be written, considering Eq. (42),
(bxbybz)=1R(S0ky0SKxkyKxS)(fxfyfz),
(54)
and with the transversality condition, Eq. (37),
0=Kxfx+kyfyfx=kyKxfy,
(55)
the following equation is satisfied
(bybz)(βΞΛβ)(fyfz).
(56)
We have made the following definitions:
βSR,ΞKxR,Λ1R(ky2Kx+Kx)=K2RKx,
(57)
with Kx being k0x, Kx or Kx for Λ, Λ′ or Λ″ (Ξ, Ξ′ or Ξ″). The continuity of the field components which are parallel to the surface (plane x = 0) can be written:
(bybyfzfzbzbzfyfy)=(b0yf0zb0zf0y)
(58)
or, using Eq. (56) and taking into account that if the material media of the incident electromagnetic wave zone is considered the vacuum, then β″ = 0 = β,
(ΞΞ0β11000βΛΛ0011)(fzfzfyfy)=(Ξ0f0zf0zΛ0f0yf0y),
(59)
where obviously, Ξ′(Λ′) and Ξ″(Λ″) are the corresponding Ξ-parameters for the transmitted and reflected waves. In addition, if one considers that Ξ″ = −Ξ0, Λ″ = −Λ0, removes for convenience the subscript 0, and recalls that fy = 0 (i.e., f perpendicular to the plane of incidence), then the result is:
(ΞΞ0β11000βΛΛ0011)(fzfzfyfy)=(Ξfzfz00).
(60)
The solutions are
fy=fy,fy=βΛ+Λfz,
(61)
and
fz=2Ξ(Λ+Λ)(Ξ+Ξ)(Λ+Λ)+β2fz,fz=(ΞΞ)(Λ+Λ)β2(Ξ+Ξ)(Λ+Λ)+β2fz.
(62)

The general relations of the components of the f-vector in function of the wave vector components are the following:
fzfz=2Rkx(Rk2/kx+RK2/Kx)(Rkx+RKx)(Rk2/kx+RK2/Kx)R2S2,fzfz=(RkxRKx)(Rk2/kx+RK2/Kx)+R2S2(Rkx+RKx)(Rk2/kx+RK2/Kx)R2S2.
(63)

These expressions are the signature of the existence of magnetic charges coexisting with electric charges, since they correspond to the Fresnel formulae when there are several kinds of dyons and therefore are different from those of the standard electromagnetism.

In the particular case S′ ≠ 0 and the incidence is perpendicular to the surface, the result is:
fyfz=βΛ+Λ=RSRk+RK,
(64)
and
fyfz=fyfzfzfz=2βΞ(ΞΞ)(Λ+Λ)β2.
(65)
There is a rotation of an angle α between the polarization plane of the incident and the reflected wave. This angle, that depends on S′, can have any value which constitutes a novelty with respect to standard electromagnetism in non birefringent materials
α=tan1(2RRSk(RkRK)(Rk+RK)+R2S2).
(66)

It is possible that a change in the polarization of the reflected light can also exist in anisotropic and other birefringent materials. However, this change depends on the polarization direction of the incident light and this dependence does not exist in the materials with magnetic charges. Therefore, the distinction between the rotation angle of the reflected wave in these birefringent compounds and that angle coming from the magnetic charges can be established by means of the above expression, Eq. (66). On the other hand, the polarization of transmitted light in the optical active materials depends on the thickness of the sample, just the opposite to the cases of materials with magnetic monopoles, whose transmitted light polarization does not depend on the thickness of their corresponding samples.

Diagonal propagation matrix

If S′ = T′ = 0, then the electric charges can coexist with magnetic charges but there are not charges with mixed charges, i.e., dyons outside of the axes of the dyonic complex plane, and then we have the following relationships between the electric fields of the different waves, the incident, transmitted and reflected electromagnetic waves:
fzfz=2kx/Rkx/R+Kx/R,fzfz=kx/RKx/Rkx/R+Kx/R.
(67)
The coefficient fz″ / fz will be real if Kx/R′ is real and then the incident and the reflected wave will have the same phase or 180 degrees different. Assuming that
KxR=1R(K2k2sinθ)1/2=R*|R|2(URk2sinθ)1/2,
(68)
then fz / fz will be real if
D2|R|2UR*(R*)2k2sinθ0.
(69)
If in a medium with electric charges, one adds magnetic monopoles and does not consider kinetic susceptibility, then the propagation matrix, Eq. (A. 3), is:
(UTSR)=cμ0(ωɛ0ɛr+iσ00ωɛ0ɛ¯r+iσ¯),
(70)
where σ is the electron conductivity and σ̄ is the monopole conductivity. Then
ImD2(μ0c)2=|R|2ωɛ0(σɛ¯rσ¯ɛr)+2ωɛ0ɛ¯rσ¯k2sinθ.
(71)
If magnetic monopoles are not present, this imaginary part never vanishes for a conductive medium, but with magnetic monopoles, this imaginary part vanishes if the following condition is satisfied
sinθ=ɛrɛ¯r2(1+σ¯2ω2ɛ02ɛ¯r2)(1σɛ¯rσ¯ɛr).
(72)
This equality can be possible for certain value of the parameters (frequency, conductivity, dielectric constant, etc.). Therefore, without monopoles there is always a phase change of the reflected wave, but with monopoles there may not be a phase change for a given angle. Thus the measurement of the phase of the reflected wave can be used to detect magnetic monopoles in material media.

In a non-conductive material without monopoles
kxR=ckcosθω(1+χm)=(1+χe1+χm)1/2cosθ=Z0Zcosθ,
(73)
where [(1 +χe)(1 + χm)]1/2 is the refraction index and Z ≡ (μ /ɛ)1/2 the impedance of the medium, and then
fzfz=2ZcosθZcosθ+Zcosθ,fzfz=ZcosθZcosθZcosθ+Zcosθ.
(74)
These relationships about transmitted and reflected fields constitute the standard Fresnel coefficients in a standard non conductive medium (see for instance Fowles [19

19. G.R. Fowles, Introduction to Modern Optics, (Dover Publications, Inc., 1975).

]).

4.2. TM polarization

Now consider that b has only z-component, bz, i.e., b is perpendicular to the plane of incidence, and parallel to the plane of separation between the vacuum and the medium. It is called p-polarization or transverse magnetic (TM) polarization.

From the curl equation (38), for a plane wave
K×b=Uf+Tbf=1U(TK×)b
(75)
that can be written, considering (42)
(fxfyfz)=1U(T0ky0TKxkyKxT)(bxbybz).
(76)
With the transversality condition, Eq. (37),
0=Kxbx+kybybx=kyKxby,
(77)
the above equation reads
(fyfz)=(βΞΛβ)(bybz),
(78)
where we define
βTU,ΞKxU,ΛK2UKx.
(79)
It should be noted that they are the same equations as in TE polarization but changing fy,z for by,z and vice versa (the value of the constants β, Ξ and Λ are also different).

The continuity of the field components parallel to the surface (x = 0 plane) can be written:
(fyfybzbzfzfzbyby)=(f0yb0zf0zb0y),
(80)
and the solutions are (identically to TE)
by=by,by=βΛ+Λbz,
(81)
and
bz=2Ξ(Λ+Λ)(Ξ+Ξ)(Λ+Λ)+β2bz,bz=(ΞΞ)(Λ+Λ)β2(Ξ+Ξ)(Λ+Λ)+β2bz.
(82)
Substituting the wave vector components
bzbz=2Ukx(Uk2/kx+UK2/Kx)(Ukx+UKx)(Uk2/kx+UK2/Kx)U2T2,bzbz=(UkxUKx)(Uk2/kx+UK2/Kx)+U2T2(Ukx+UKx)(Uk2/kx+UK2/Kx)U2T2.
(83)
Equations (83) corresponding to the magnetic field are equivalent to those of the electric one [Eqs. (63)]. Therefore, the arguments used in these equations (63) can be extended to these latter results.

The establishment of these relationships between the transmitted and reflected b-field components along with those of Eqs. (63) constitute one of the main effective results of this paper, since they are a distinctive signature of the existence of magnetic monopoles.

Diagonal propagation matrix

If S′ = T′ = 0 then
bz=2UkxUkx+UKxbz,bz=UkxUKxUkx+UKxbz.
(84)

Without magnetic monopoles the U value is
U=μ0c(ωɛ+iσ)
(85)
and then, in a non conductive material
kxU=kcosθμ0cωɛ=(ɛrμr)1/2cosθɛr=ZZ0cosθ,
(86)
where Z ≡ (μ/ɛ)1/2 the impedance of the medium. They are also the standard Fresnel coefficients in a non-conductive medium:
bzbz=2ZcosθZcosθ+Zcosθ,bzbz=ZcosθZcosθZcosθ+Zcosθ.
(87)

If we want the Fresnel coefficients depending on f instead of b, we consider Eq. (76) and
|f|2fx2+fy2+fz2=1U2(ky2+Kx2)bz2=K2U2bz2,
(88)
therefore
|f||f|=KkUUbzbz=ZZbzbz.
(89)
The Fresnel coefficients read
|f||f|=2ZcosθZcosθ+Zcosθ,|f||f|=ZcosθZcosθZcosθ+Zcosθ.
(90)

5. Summary and conclusions

From Maxwell’s equations in material media containing magnetic monopoles, we have studied the propagation of electromagnetic waves in the media. The wave vector for a plane wave can be complex even if the medium is non-conductive, Eq. (45). This is an important difference compared to the media without monopoles, since a non-conductive electric compound with magnetic monopoles presents attenuation of the electromagnetic wave. This never happens without magnetic monopoles. However, this wave vector is reduced to the standard if there are no magnetic monopoles.

We have also calculated the reflected and transmitted waves when an electromagnetic field propagates from the vacuum to a medium. This is carried out taking into account the two possible polarizations, the electric field in the vacuum perpendicular to the plane of incidence (TE polarization) and the electric field parallel to the plane of incidence (TM polarization). The reflected wave can be used as proof of experimental existence of magnetic monopoles (for example in spin-ice) because its characteristics are modified whether or not such monopoles are present. From Eqs. (47), (48) and (49), we show the total symmetry between magnetic and electric conductivities. This symmetry is such that an electromagnetic wave penetrating in an electric insulator with magnetic monopoles behaves identically with another electromagnetic wave penetrating in an electric conductor without magnetic monopoles. These symmetrical behaviors show as unique quantitative differences those that depend on the values of the respective conductivities. If there are dyons, particles with ζ ≠ 0, π/2, even the plane of polarization of the reflected wave would be modified with respect to the polarization of the incident wave. In any case, we want to emphasize that the most obvious change is the phase of the reflected wave, since, in conducting material media with monopoles, the reflected wave may have the same phase as the incident, situation that never occurs without existence of magnetic monopoles.

Finally, we want to emphasize, as written in our introduction, that the objective of this paper was to carry out a formalization of the electromagnetic wave propagation in semi-infinite media with electric charges and magnetic monopoles, having in mind the arguments used in the standard electromagnetism. The results we obtain with this theory are a consequence of the extended Maxwell equations formulated within the matter. However, the 2009–2010 experiments are compatible with our theoretical construction. This compatibility can give a possible legitimacy to our argumentation line in such a way that the other properties deduced from our results may coincide with those properties that can be obtained in future experiments carried out in materials with magnetic charges. These new future experimental data will serve us for either rectifying or ratifying the starting points of our analysis.

A. Propagation matrix with two classes of particles

In this Appendix, we calculate the propagation matrix with two classes of dyons, i.e., there exist charges with different phases, ζ and ζ̄, then the permeability matrix (33) is
μ=μ01ξξ+ξξsin2(ζζ)(𝟙ξΘξ¯Θ¯),
(A. 1)
and the propagation matrix (39) is written as
(UTSR)=c(ωɛ+iσ)μ=c[ωɛ0(𝟙+χΘ+χ¯Θ¯)+iσΘ+iσ¯Θ¯]μ01ξξ+ξξsin2(ζζ)(𝟙ξΘξ¯Θ¯).
(A. 2)
It should be remarked that this matrix is, in general a non-symmetric operator, (ST), because ΘΘ̄ is non-symmetric.

In the case that ζ ζ̄ = ±π/2 (from a duality transformation, there are only electric charges and magnetic monopoles, but there are not dyons with both electric and magnetic charges), then, ΘΘ̄ = 0 and sin2(ζ ζ̄) = 1, and the propagation matrix is:
(UTSR)=cμ0ωɛ0(1ξ)(1ξ)[𝟙+(χξχξ)Θ+(χ¯ξ¯χ¯ξ¯)Θ¯++iσ(1ξ)ωɛ0Θ+iσ¯(1ξ¯)ωɛ0Θ¯],
(A. 3)
that is a symmetric matrix (S = T), and consequently
K2=RUTS=ɛ0(1+χ)(1+χ¯)μ0(1ξ)(1ξ)ω2[1+iσωɛ0(1+χ)][1+iσ¯ωɛ0(1+χ¯)]=ɛ0ɛrɛ¯rμ0μrμ¯rω2[1+iσωɛ0ɛr][1+iσ¯ωɛ0ɛ¯r]=ɛμω2[1σσ¯ω2ɛ0ɛ+iωɛ0(σɛr+σ¯ɛ¯r)].
(A. 4)
This result becomes the standard value of K for a conductor medium, when σ̄ = ξ̄ = 0, i.e., when there are only electric charges and the corresponding electric conductivity σ.

References and links

1.

P. Peregrinus, The Letter of Petrus Peregrinus, translated by B. Arnold (McGraw Publishing Company, 1904).

2.

P. A. M. Dirac, “Quantized Singularities in the Electromagnetic Field,” Proc. R. Soc. Lond. A133, 60 (1931).

3.

P. A. M. Dirac, “The Theory of Magnetic Poles,” Phys. Rev. 74, 817–830 (1948). [CrossRef]

4.

G. ’t Hooft, “Magnetic monopoles in unified gauge theories,” Nucl. Phys. B79, 276–284 (1974). [CrossRef]

5.

B. Cabrera, “First Results from a Superconductive Detector for Moving Magnetic Monopoles,” Phys. Rev. Lett. 48, 1378–1381 (1982). [CrossRef]

6.

C. Castelnovo, R. Moessner, and S. L. Sondhi, “Magnetic monopoles in spin ice,” Nature 451, 42–45 (2008). [CrossRef] [PubMed]

7.

X.-L. Qi, R. Li, J. Zang, and S.-C. Zhang, “Inducing a Magnetic Monopole with Topological Surface States,” Science 323, 1184–1187 (2009). [CrossRef] [PubMed]

8.

S. T. Bramwell, S. R. Giblin, S. Calder, R. Aldus, D. Prabhakaran, and T. Fennell, “Measurement of the charge and current of magnetic monopoles in spin ice,” Nature 461, 956–960 (2009). [CrossRef] [PubMed]

9.

D. J. P. Morris, D. A. Tennant, S. A. Grigera, B. Klemke, C. Castelnovo, R. Moessner, C. Czternasty, M. Meissner, K. C. Rule, J.-U. Hoffmann, K. Kiefer, S. Gerischer, D. Slobinsky, and R. S. Perry, “Dirac Strings and Magnetic Monopoles in the Spin Ice Dy2Ti2O7,” Science 326, 411–414 (2009). [CrossRef] [PubMed]

10.

M. J. Gingrass, “Observing monopoles in a magnetic analog of ice,” Science 326, 375–376 (2009). [CrossRef]

11.

S. Ladak, D. E. Read, G. K. Perkins, L. F. Cohen, and W. R. Branford, “Direct observation of magnetic monopole defects in an artificial spin-ice system,” Nat. Phys. 6, 359–363 (2010). [CrossRef]

12.

S. Sondhi, “Wien route to monopoles,” Nature 461, 888–889 (2009). [CrossRef] [PubMed]

13.

L. J. Onsager, “Deviations from Ohm’s Law in Weak Electrolytes,” J. Chem. Phys. 2, 599–615 (1934). [CrossRef]

14.

See for example, J. D. Jackson, Classical Electrodynamics, third edition (John Wiley & Sons, Inc., 1999).

15.

K. A. Milton, “Theoretical and experimental status of magnetic monopoles,” Rep. Prog. Phys. 69, 1637–1711 (2006). [CrossRef]

16.

J. Costa-Quintana and F. Lopez-Aguilar, “Extended classical electrodynamics with magnetic monopoles,” Far East J. Mech. Eng. Phys. 1, 19–56 (2010).

17.

J. Costa-Quintana and F. Lopez-Aguilar, “Propagation of electromagnetic waves in material media with magnetic monopoles,” Prog. Electromagn. Res. 110, 267–295 (2010). [CrossRef]

18.

Y.M. Shnir, Magnetic monopoles, (Springer-Verlag, 2005).

19.

G.R. Fowles, Introduction to Modern Optics, (Dover Publications, Inc., 1975).

OCIS Codes
(160.3820) Materials : Magneto-optical materials
(260.2110) Physical optics : Electromagnetic optics

ToC Category:
Physical Optics

History
Original Manuscript: November 29, 2010
Revised Manuscript: February 3, 2011
Manuscript Accepted: February 6, 2011
Published: February 11, 2011

Citation
J. Costa-Quintana and F. López-Aguilar, "Fresnel coefficients in materials with magnetic monopoles," Opt. Express 19, 3742-3757 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-4-3742


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References

  1. P. Peregrinus, The Letter of Petrus Peregrinus, translated by B. Arnold (McGraw Publishing Company, 1904).
  2. P. A. M. Dirac, "Quantized Singularities in the Electromagnetic Field," Proc. R. Soc. Lond. 133, 60 (1931).
  3. P. A. M. Dirac, "The Theory of Magnetic Poles," Phys. Rev. 74, 817-830 (1948). [CrossRef]
  4. G. ’t Hooft, "Magnetic monopoles in unified gauge theories," Nucl. Phys. B 79, 276-284 (1974). [CrossRef]
  5. B. Cabrera, "First Results from a Superconductive Detector for Moving Magnetic Monopoles," Phys. Rev. Lett. 48, 1378-1381 (1982). [CrossRef]
  6. C. Castelnovo, R. Moessner, and S. L. Sondhi, "Magnetic monopoles in spin ice," Nature 451, 42-45 (2008). [CrossRef] [PubMed]
  7. X.-L. Qi, R. Li, J. Zang, and S.-C. Zhang, "Inducing a Magnetic Monopole with Topological Surface States," Science 323, 1184-1187 (2009). [CrossRef] [PubMed]
  8. S. T. Bramwell, S. R. Giblin, S. Calder, R. Aldus, D. Prabhakaran, and T. Fennell, "Measurement of the charge and current of magnetic monopoles in spin ice," Nature 461, 956-960 (2009). [CrossRef] [PubMed]
  9. D. J. P. Morris, D. A. Tennant, S. A. Grigera, B. Klemke, C. Castelnovo, R. Moessner, C. Czternasty, M. Meissner, K. C. Rule, J.-U. Hoffmann, K. Kiefer, S. Gerischer, D. Slobinsky, and R. S. Perry, "Dirac Strings and Magnetic Monopoles in the Spin Ice Dy2Ti2O7," Science 326, 411-414 (2009). [CrossRef] [PubMed]
  10. M. J. Gingrass, "Observing monopoles in a magnetic analog of ice," Science 326, 375-376 (2009). [CrossRef]
  11. S. Ladak, D. E. Read, G. K. Perkins, L. F. Cohen, and W. R. Branford, "Direct observation of magnetic monopole defects in an artificial spin-ice system," Nat. Phys. 6, 359-363 (2010). [CrossRef]
  12. S. Sondhi, "Wien route to monopoles," Nature 461, 888-889 (2009). [CrossRef] [PubMed]
  13. L. J. Onsager, "Deviations from Ohm’s Law in Weak Electrolytes," J. Chem. Phys. 2, 599-615 (1934). [CrossRef]
  14. See for example,J. D. Jackson, Classical Electrodynamics, third edition (John Wiley & Sons, Inc., 1999).
  15. K. A. Milton, "Theoretical and experimental status of magnetic monopoles," Rep. Prog. Phys. 69, 1637-1711 (2006). [CrossRef]
  16. J. Costa-Quintana, and F. Lopez-Aguilar, "Extended classical electrodynamics with magnetic monopoles," Far East J. Mech. Eng. Phys. 1, 19-56 (2010).
  17. J. Costa-Quintana, and F. Lopez-Aguilar, "Propagation of electromagnetic waves in material media with magnetic monopoles," Prog. Electromagn. Res. 110, 267-295 (2010). [CrossRef]
  18. Y. M. Shnir, Magnetic monopoles, (Springer-Verlag, 2005).
  19. G. R. Fowles, Introduction to Modern Optics, (Dover Publications, Inc., 1975).

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